06- computational fluid dynamics and heat transfer emerging topics

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International Series on Developments in Heat Transfer

Objectives

The Developments in Heat Transfer book Series publishes state-of-the-art booksand provides valuable contributions to the literature in the field of heat transfer,thermal and energy engineering. The overall aim of the Series is to bring to theattention of the international community recent advances in thermal sciences byauthors in academic research and the engineering industry.

Research and development in heat transfer is of significant importance to manybranches of technology, not least in energy technology. Developments include new,efficient heat exchangers, novel heat transfer equipment as well as the introductionof systems of heat exchangers in industrial processes. Application areas include heatrecovery in the chemical and process industries, and buildings and dwelling houseswhere heat transfer plays a major role. Heat exchange combined with heat storageis also a methodology for improving the energy efficiency in industry, while coolingin gas turbine systems and combustion engines is another important area of heattransfer research. Emerging technologies like fuel cells and batteries also involvesignificant heat transfer issues.

To progress developments within the field both basic and applied research isneeded. Advances in numerical solution methods of partial differential equations,turbulence modelling, high-speed, efficient and cheap computers, advancedexperimental methods using LDV (laser-doppler-velocimetry), PIV (particle-image-velocimetry) and image processing of thermal pictures of liquid crystals, have all ledto dramatic advances during recent years in the solution and investigation ofcomplex problems within the field.

The aims of the Series are achieved by contributions to the volumes from invitedauthors only. This is backed by an internationally recognised Editorial Board for theSeries who represent much of the active research worldwide. Volumes planned forthe series include the following topics: Compact Heat Exchangers, EngineeringHeat Transfer Phenomena, Fins and Fin Systems, Condensation, Materials Processing,Gas Turbine Cooling, Electronics Cooling, Combustion-Related Heat Transfer,Heat Transfer in Gas-Solid Flows, Thermal Radiation, the Boundary ElementMethod in Heat Transfer, Phase Change Problems, Heat Transfer in Micro-Devices,Plate-and-Frame Heat Exchangers, Turbulent Convective Heat Transfer in Ducts,Enhancement of Heat Transfer, Transport Phenomena in Fires, Fuel Cells andBatteries as well as Thermal Issues in Future Vehicles and other selected topics.

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Associate Editors

R. AmanoUniversity of Wisconsin, USA

C.A. BrebbiaWessex Institute of Technology, UK

G. CominiUniversity of Udine, Italy

R.M. CottaCOPPE/UFRJ, Brazil

S.K. DasIndia Institute of Technology, Madras,India

L. De BiaseUniversity of Milan, Italy

G. De MeyUniversity of Ghent, Belgium

S. del GuidiceUniversity of Udine, Italy

M. FaghriUniversity of Rhode Island, USA

C. HermanJohn Hopkins University, USA

Y. JaluriaRutgers University, USA

S. KabelacHelmut Schmidt University, Hamburg,Germany

D.B. MurrayTrinity College Dublin, Ireland

P.H. OosthuizenQueen’s University Kingston, Canada

P. PoskasLithuanian Energy Institute, Lithuania

B. SarlerNova Gorica Polytechnic, Slovenia

A.C.M. SousaUniversity of New Brunswick, Canada

D.B. SpaldingCHAM, UK

J. SzmydUniversity of Mining and Metallurgy,Poland

D. TaftiViginia Tech., USA

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CCCCComputational Fluid Dynamicsomputational Fluid Dynamicsomputational Fluid Dynamicsomputational Fluid Dynamicsomputational Fluid Dynamicsand Heat Tand Heat Tand Heat Tand Heat Tand Heat Transfransfransfransfransfererererer

EMERGING TOPICS

Editors

R.S. AmanoUniversity of Wisconsin-Milwaukee, USA

&

B. SundénLund University, Sweden

Published by

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R.S. AmanoUniversity of Wisconsin-Milwaukee, USA

B. SundénLund University, Sweden

Contents

Preface xiii I. Finite-Volume Method 1

1 A higher-order bounded discretization scheme 3 Baojun Song and R.S. Amano

1.1 Introduction …………………………………………………… 3 1.2 Numerical Formulation ..............................................................6 1.2.1 Governing equations.......................................................6 1.2.2 Discretization..................................................................6 1.2.3 Higher-order schemes.....................................................7 1.2.4 Weighted-average coefficient ensuring

boundedness ...................................................................8 1.3 Test Problem and Results .........................................................11 1.3.1 Pure convection of a box-shaped step profile ...............11

1.3.2 Sudden expansion of an oblique velocity field in a cavity............................................................................12

1.3.3 Two-dimensional laminar flow over a fence ................15 1.4 Conclusions ..............................................................................16 2 Higher-order numerical schemes for heat, mass, and

momentum transfer in fluid flow 19 Mohsen M.M. Abou-Ellail, Yuan Li and Timothy W. Tong

2.1 Introduction ..............................................................................20 2.2 Single-Grid Schemes ................................................................21 2.3 New Numerical Simulation Strategy........................................22 2.4 Novel Multigrid Numerical Procedure .....................................23 2.5 The First Test Problem……………………………………….. 29 2.6 Numerical Results of the First Test Problem............................30 2.7 The Second Test Problem……………………………………. 39 2.8 Numerical Results of the Second Test Problem .......................41 2.9

Application of NIMO Scheme to Laminar Flow Problems...................................................................................44

2.9.1 Steady laminar flow in pipes ........................................47 2.9.2 Steady laminar flow over a fence…………………… 48

2.10

Application of the NIMO Higher-Order Scheme to the Turbulent Flow in Pipes ...........................................................52

2.11 Conclusions ..............................................................................58 3 CFD for industrial turbomachinery designs 61 C. Xu and R.S. Amano

3.1 Introduction ..............................................................................61 3.1.1 Computational methods in turbomachinery..................61 3.1.2 Grid-free vortex method ...............................................63 3.2 Numerical Methods for Incompressible Flow ..........................64 3.3 Numerical Methods for Compressible Flow.............................65 3.4 Governing Equations for Two-Dimensional Flow ...................70 3.5 Decomposition of Flux Vector .................................................72 3.5.1 Governing equations.....................................................72 3.5.2 Special treatment of the artificial dissipation terms

and numerical algorithm..............................................74 3.6 Stability Analysis .....................................................................77 3.7 Applications in Turbine Cascade..............................................80 3.7.1 C3X turbine cascade.....................................................80 3.7.2 VKI turbine cascade .....................................................85 3.8 Numerical Method for Three-Dimensional Flows....................88 3.9 Applications of Three-Dimensional Method ............................89

3.9.1 Analysis of pitch-width effects on the secondaryflows of turbine blades ................................................89

3.9.2 Flow around centrifuge compressors scroll tongue ....101 3.10 CFD Applications in Turbomachine Design ..........................113 3.10.1 Flow solver for section analysis ...............................116 3.10.2 Optimization .............................................................116

II. Finite Element Method 127

4 The finite element method: discertization and application to heat convection problems 129

Alessandro Mauro, Perumal Nithiarasu, Nicola Massarotti and Fausto Arpino

4.1 Governing Equations ..............................................................129 4.1.1 Non-dimensional form of fluid flow equations ..........129 4.1.2 Non-dimensional form of turbulent flow equations....138 4.1.3 Porous media flow: the generalized model equations.143 4.2 The Finite Element Method....................................................146 4.2.1 Strong and weak forms ...............................................146 4.2.2 Weighted residual approximation...............................148 4.2.3 The Galerkin, finite element, method .........................149

4.2.4 Characteristic Galerkin scheme for convection-diffusion equation...................................150

4.2.5 Stability conditions .................................................... 155 4.2.6 Characteristic-based split scheme .............................. 157 5 Equal-order segregated finite-element method

for fluid flow and heat transfer simulation 171 Jianhui Xie

5.1 Introduction............................................................................ 171 5.2 Finite-Element Description.................................................... 175 5.2.1 Two-dimensional elements ........................................ 175 5.2.2 Three-dimensional elements ...................................... 177 5.2.3 Degenerated elements ................................................ 179 5.2.4 Special elements (rod and shell) ................................ 179 5.3

Governing Equations for Fluid Flow and Heat Transfer Problems ................................................................................ 179

5.3.1 General form of governing equations ........................ 180 5.3.2 Discretized equations and solution algorithm ............ 182 5.3.3 Stabilized method....................................................... 187 5.4 Formulation of Stabilized Equal-Order Segregated Scheme . 193 5.4.1 Introduction................................................................ 193 5.4.2 FEM-based segregated formulation ........................... 195 5.4.3 Data storage and block I/O process............................ 205 5.5 Case Studies........................................................................... 208 5.5.1 Two-dimensional air cooling box .............................. 208 5.5.2 CPU water cooling analysis ....................................... 210

III.

Turbulent Flow Computations/Large Eddy Simulation/Direct Numerical Simulation 215

6

Time-accurate techniques for turbulent heat transfer analysis in complex geometries 217

Danesh K. Tafti

6.1 Introduction............................................................................ 217 6.2 General Form of Conservative Equations .............................. 218 6.2.1 Incompressible constant property assumption ........... 220 6.2.2 Modeling turbulence .................................................. 224 6.3

Transformed Equations in Generalized Coordinate Systems ............................................................... 225

6.3.1 Source terms in rotating systems................................ 226 6.4 Computational Framework .................................................... 227 6.5 Time-Integration Algorithm................................................... 231 6.5.1 Predictor step ............................................................. 231 6.5.2 Pressure formulation and corrector step..................... 232 6.5.3 Integral adjustments at nonmatching boundaries ....... 239

6.6 Discretization of Convection Terms.......................................240 6.7 Large-Eddy Simulations and Subgrid Modeling ....................246 6.8 Detached Eddy Simulations or Hybrid RANS-LES...............252 6.9 Solution of Linear Systems ....................................................254 6.10 Parallelization Strategies ........................................................256 6.11 Applications ...........................................................................259 7

On large eddy simulation of turbulent flow and heat transfer in ribbed ducts 265

Bengt Sundén and Rongguang Jia

7.1 Introduction ............................................................................265 7.2 Numerical Method..................................................................267 7.3 Results And Discussion..........................................................267 7.3.1 Fully developed pipe flow ..........................................267 7.3.2 Transverse ribbed duct flow .......................................269 7.3.3 V-shaped ribbed duct flow .........................................271 7.4 Conclusions ............................................................................273 8

Recent developments in DNS and modeling of turbulent clows and heat transfer 275

Yasutaka Nagano and Hirofumi Hattori

8.1 Introduction ............................................................................275 8.2 Present State of Direct Numerical Simulations ......................276 8.3 Instantaneous and Reynolds-Averaged Governing Equations for Flow and Heat Transfer ...................................277 8.4 Numerical Procedures of DNS ...............................................279 8.4.1 DNS using high-accuracy finite-difference method ...280 8.4.2 DNS using spectral method ........................................280 8.5 DNS of Turbulent heat Transfer in Channel Flow with Transverse-Rib Roughness: Finite-Difference Method..........281 8.5.1 Heat transfer and skin friction coefficients.................281 8.5.2 Velocity and thermal fields around the rib .................284

8.5.3 Statistical characteristics of velocity field and turbulent structures....................................................287

8.5.4 Statistical characteristics of thermal field and related turbulent structures....................................................291

8.6

DNS of Turbulent heat Transfer in Channel Flow with Arbitrary Rotating Axes: Spectral Method.............................295

8.7

Nonlinear Eddy Diffusivity Model for Wall-Bounded Turbulent Flow.......................................................................301

8.7.1 Evaluations of existing turbulence models in rotating wall-bounded flows......................................301

8.7.2 Proposal of nonlinear eddy diffusivity model for wall-bounded flow.....................................................304

8.8 Nonlinear Eddy Diffusivity Model for Wall-Bounded Turbulent Heat Transfer......................................................... 310

8.8.1 Evaluations of turbulent heat transfer models in rotating channel flows............................................... 310

8.8.2 Proposal of nonlinear eddy diffusivity model for wall-bounded turbulent heat transfer......................... 312

8.9 Model Performances .............................................................. 316 8.9.1 Prediction of rotating channel flow using NLEDMM 316

8.9.2 Prediction of rotating channel-flow heat transfer using NLEDHM........................................................ 322

8.10 Concluding Remarks.............................................................. 327 9

Analytical wall-functions of turbulence for complex surface flow phenomena 331

K. Suga

9.1 Introduction............................................................................ 331 9.2 Numerical Implementation of Wall Functions....................... 333 9.3 Standard Log-Law Wall-Function (LWF) ............................. 334 9.4 Analytical Wall-Function (AWF) .......................................... 335 9.4.1 Basic strategy of the AWF ......................................... 335 9.4.2 AWF in non-orthogonal grid systems ........................ 338

9.4.3 AWF for rough wall turbulent flow and heat transfer ....................................................................... 339

9.4.4 AWF for permeable walls .......................................... 352 9.4.5 AWF for high Prandtl number flows.......................... 359 9.4.6 AWF for high Schmidt number flows........................ 365 9.5 Conclusions............................................................................ 369 9.6 Nomenclature......................................................................... 376

IV. Advanced Simulation Modeling Technologies 381 10 SPH – a versatile multiphysics modeling tool 383 Fangming Jiang and Antonio C.M. Sousa

10.1 Introduction............................................................................ 383 10.2 SPH Theory, Formulation, and Benchmarking...................... 384 10.2.1 SPH theory and formulation...................................... 384 10.2.2 Benchmarking ........................................................... 389 10.3 Control of the Onset of Turbulence in MHD Fluid Flow....... 392 10.3.1 MHD flow control..................................................... 394 10.3.2 MHD modeling ......................................................... 395

10.3.3 SPH analysis of magnetic conditions to restrain the transition to turbulence ............................................. 396

10.4

SPH Numerical Modeling for Ballistic-Diffusive Heat Conduction .................................................................... 400

10.4.1 Transient heat conduction across thin films ............. 402

10.4.2 SPH modeling .......................................................... 403 10.4.3 Boundary treatment .................................................. 405 10.4.4 Results ...................................................................... 406 10.5

Mesoscopic Pore-Scale SPH Model for Fluid Flow in Porous Media ..................................................................... 409

10.5.1 Modeling strategy..................................................... 412 10.5.2 Results and discussion.............................................. 416 10.6 Concluding Remarks.............................................................. 420 11 Evaluation of continuous and discrete phase models

for simulating submicrometer aerosol transport and deposition 425

Philip Worth Longest and Jinxiang Xi

11.1 Introduction............................................................................ 426 11.2 Models of Airflow and Submicrometer Particle Transport .... 428 11.2.1 Chemical species model for particle transport ......... 428 11.2.2 Discrete phase model ............................................... 429 11.2.3 Deposition factors .................................................... 430 11.2.4 Numerical methods .................................................. 430 11.3 Evaluation of Inertial Effects on Submicrometer Aerosols.... 431 11.4 An Effective Eulerian-Based Model for Simulating Submicrometer Aerosols………………………………… 434 11.5

Evaluation of the DF-VC Model in an Idealized Airway Geometry................................................................... 437

11.6 Evaluation of the DF-VC Model in Realistic Airways........... 441 11.6.1 Tracheobronchial region .......................................... 441 11.6.2 Nasal cavity.............................................................. 446 11.7 Discussion .............................................................................. 451 12

Algorithm stabilization and acceleration in computational fluid dynamics: exploiting recursive properties of fixed point algorithms 459

Aleksandar Jemcov and Joseph P. Maruszewski

12.1 Introduction............................................................................ 460 12.2 Iterative Methods for Flow Equations.................................... 462 12.2.1 Discrete form of governing equations ...................... 463 12.2.2 Recursive property of iterative methods................... 465 12.3 Reduced Rank Extrapolation.................................................. 467 12.4 Numerical Experiments.......................................................... 470 12.4.1 RRE acceleration of implicit density-based solver... 470 12.4.2 RRE acceleration of explicit density-based solver ... 475

12.4.3 RRE acceleration of segregated pressure-based solver ........................................................................ 477

12.4.4 RRE acceleration of coupled pressure-based solver. 480 12.5 Conclusion ............................................................................. 482

Sunden Prelims.tex 15/9/2010 15: 48 Page vii

Preface

The main focus of this book is to introduce computational methods for fluid flowand heat transfer to scientists, engineers, educators, and graduate students whoare engaged in developing and/or using computer codes. The topic ranges frombasic methods such as a finite difference, finite volume, finite element, large-eddy simulation (LES), and direct numerical simulation (DNS) to advanced, andsmoothed particle hydrodynamics (SPH). The objective is to present the currentstate-of-the-art for simulating fluid flow and heat transfer phenomena in engineeringapplications.

The first and second chapters present higher-order numerical schemes. Theseschemes include second-order UPWIND, QUICK, weighted-average coefficientensuring boundedness (WACEB), and non-upwind interconnected multigrid over-lapping (NIMO) finite-differencing and finite-volume methods. Chapter 3 givesoverview of the finite-difference and finite-volume methods covering subsonic tosupersonic flow computations, numerical stability analysis, eigenvalue-stiffnessproblem, features of two- and three-dimensional computational schemes, and flux-vector splitting technique. The chapter shows a few case studies for gas turbineblade design and centrifugal compressor flow computations.

The fourth and fifth chapters give overview of the finite-element methodand its applications to heat and fluid flow problems. An introduction toweighted residual approximation and finite-element method for heat and fluidflow equations are presented along with the characteristic-based split algorithmin Chapter 4. Chapter 5 discusses two important concepts. One is the equal-order mixed-GLS (Galerkin Least Squares) stabilized formulation, which is ageneralization of SUPG (streamline-upwind/Petrov–Galerkin) and PSPG (pressurestabilizing/Petrov–Galerkin) method. The second is the numerical strategies for thesolution of large systems of equations arising from the finite-element discretizationof the above formulations. To solve the nonlinear fluid flow/heat transfer problem,particular emphasis is placed on segregated scheme (SIMPLE like in finite-volumemethod) in nonlinear level and iterative methods in linear level.

Chapters 6 through eight give numerical methods to solve turbulent flows. Chap-ter 6 overviews most important methods including RANS approach, LES, and DNSand describes the numerical and theoretical background comprehensively to enablethe use of these methods in complex geometries. Chapter 7 demonstrates the advan-tages of large-eddy simulation (LES) for computations of the flow and heat transferin ribbed ducts through a gas turbine blade. Direct numerical simulation (DNS)

Sunden Prelims.tex 15/9/2010 15: 48 Page viii

is introduced in Chapter 8. In this chapter recent studies on DNS and turbulencemodels from the standpoint of computational fluid dynamics (CFD) and compu-tational heat transfer (CHT) are reviewed and the trends in recent DNS researchand its role in turbulence modeling are discussed in detail. Chapter 9 discussesthe analytical wall-function of turbulence for complex surface flows. This chap-ter introduces the recently emerged analytical wall-function (AWF) methods forsurface boundary conditions of turbulent flows.

Some advanced simulation modeling technologies are given in chapters 10 and11. In Chapter 10 the current state-of-the-art and recent advances of a novel numer-ical method – the smoothed particle hydrodynamics (SPH) is reviewed throughcase studies with particular emphasis on fluid flow and heat transport. To providesufficient background and to assess its engineering/scientific relevance, three par-ticular case studies are used to exemplify macro- and nanoscale applications ofthis methodology. The first application in this chapter deals with magnetohydrody-namic (MHD) turbulence control. Chapter 11 provides the continuous and discretephase models for simulating submicrometer aerosol transport and deposition. LastlyChapter 12 discusses convergence acceleration of nonlinear flow solvers throughuse of techniques that exploit recursive properties fixed-point methods of CFDalgorithms.

The authors of the chapters were all invited to contribute to this book in accor-dance with their expert knowledge and background. All of the chapters follow aunified outline and presentation to aid accessibility and the book provides invaluableinformation to researchers in computational studies.

Finally, we are grateful to the authors and reviewers for their excellent con-tributions to complete this book. We are thankful for the ceaseless help that wasprovided by the staff members of WIT Press, in particular Mr. Brian Privett andMrs. Elizabeth Cherry, and for their encouragement in the production of this book.Finally, our appreciation goes to Dr. Carlos Brebbia who gave us strong supportand encouragement to complete this project.

Ryoichi S. Amano and Bengt Sunden

Sunden CH001.tex 17/8/2010 20: 14 Page 1

I. Finite-Volume Method


1 A higher-order bounded discretizationscheme

Baojun Song and Ryo S. AmanoUniversity of Wisconsin–Milwaukee, Milwaukee, WI, USA

Abstract

This chapter presents an overview of the higher-order scheme and introduces a newhigher-order bounded scheme, weighted-average coefficient ensuring boundedness(WACEB), for approximating the convective fluxes in solving transport equationswith the finite-volume difference method.The weighted-average formulation is usedfor interpolating the variables at cell faces, and the weighted-average coefficient isdetermined from normalized variable formulation and total variation diminishing(TVD) constraints to ensure the boundedness of solutions. The new scheme is testedby solving three problems: (1) a pure convection of a box-shaped step profile in anoblique velocity field, (2) a sudden expansion of an oblique velocity field in a cavity,and (3) a laminar flow over a fence. The results obtained by the present WACEBare compared with the upwind and QUICK schemes and show that this schemehas at least the second-order accuracy while ensuring boundedness of solutions.Moreover, it is demonstrated that this scheme produces results that better agreewith the experimental data in comparison with other schemes.

Keywords: Finite-volume method, Higher-order scheme

1.1 Introduction

The approximation of the convection fluxes in the transport equations has a decisiveinfluence on the overall accuracy of any numerical solution for fluid flow and heattransfer. Although convection is represented by a simple first-order derivative, itsnumerical representation remains one of the central issues in CFD. The classic first-order schemes such as upwind, hybrid, and power-law are unconditionally bounded,but tend to misrepresent the diffusion transport process through the addition ofnumerical or “false” diffusion arising from flow-to-grid skewness. Higher-orderschemes, such as the second-order upwind [1] and the third-order upwind (QUICK)[2], offer a route to improve accuracy of the computations. However, they allsuffer from the boundedness problem; that is, the solutions may display unphysical


4 Computational Fluid Dynamics and Heat Transfer

oscillations in regions of steep gradients, which can be sufficiently serious to causenumerical instability.

During the past two decades, efforts have been made to derive higher res-olution and bounded schemes. In 1988, Zhu and Leschziner proposed a localoscillation-damping algorithm (LODA) [3]. Since the LODA scheme introducesthe contribution of the upwind scheme, the second-order diffusion is introducedinto those regions where QUICK displays unbounded behavior. In 1988, Leonard[4] developed a normalized variable formulation and presented a high-resolutionbounded scheme named SHARP (simple high-accuracy resolution program).Gaskell and Lau [5] developed a scheme called SMART (sharp and monotonic algo-rithm for realistic transport), which employs a curvature-compensated convectivetransport approximation and a piecewise linear normalized variable formulation.However, numerical testing [6] shows that both SMART and SHARP need an under-relaxation treatment at each of the control volume cell faces in order to suppressthe oscillatory convergence behavior. This drawback leads to an increase in thecomputer storage requirement, especially for three-dimensional flow calculation.In 1991, Zhu [7] proposed a hybrid linear/parabolic approximation (HLPA) scheme.However, this method has only the second-order accuracy.

In the present study, a weighted-averaged formulation is employed to interpo-late variables at cell faces and the weighted-average coefficient is determined basedon the normalized variable formulation and total variation diminishing (TVD) con-straints. Three test cases are examined: a pure convection of a box-shaped stepprofile in an oblique velocity field, a sudden expansion of an oblique velocity fieldin a cavity, and laminar flow over a fence. Computations are performed on a gener-alized curvilinear coordinate system. The schemes are implemented in a deferredcorrection approach. The computed results are compared with those obtained usingQUICK and upwind schemes and available experimental data.

In CFD research, there are three major categories to be considered for flowstudies in turbines:

1. Mathematical models – the physical behaviors that are to be predicted totallydepend on mathematical models. The choice of mathematical models shouldbe carefully made, such as inviscid or viscous analysis, turbulence models,inclusion of buoyancy, rotation, Coriolis effects, density variation, etc.

2. Numerical models – selection of a numerical technique is very important to judgewhether or not the models can be effectively and accurately solved. Factors thatneed to be reviewed for computations include the order of accuracy, treatmentof artificial viscosity, consideration of boundedness of the scheme, etc.

3. Coordinate systems – the type and structure of the grid (structured or unstruc-tured grids) directly affect the robustness of the solution and accuracy.

Numerical studies demand, besides mathematical representations of the flowmotion, a general, flexible, efficient, accurate, and – perhaps most importantly –stable and bounded (free from numerical instability) numerical algorithm for solv-ing a complete set of average equations and turbulence equations. The formulation


A higher-order bounded discretization scheme 5

Table 1.1. Schemes used in CFD

Scheme Developers Order False Boundednessdiffusion

Upwind – 1st High Unconditionallybounded

Hybrid Gosman (1977) 1st High Unconditionallybounded

Power-law Patankar (1980) 1st Medium Unconditionallybounded

Second-order Price et al. (1966) 2nd Low Unboundedupwind

QUICK Leonard (1979) 3rd Low Unbounded

LODA Zhu–Leschziner 2nd Low Conditionally(1988) bounded

SHARP Leonard (1988) 2nd Low Conditionallybounded

SMART Gaskell–Lau (1988) 2nd Low Conditionallybounded

WACEB Song et al. (1999) 2nd Low Unconditionallybounded

of the discretization scheme of convection fluxes may be one of the major tasks tomeet such demands.

As for the numerical method, the classic first-order schemes such as upwind,hybrid, and power-law [8] are unconditionally bounded (solutions do not sufferfrom over/undershoot), but tend to misrepresent the transport process throughaddition of numerical diffusion arising from flow-to-grid skewness. These are theschemes that most of the commercial codes employ. In some applications, smallovershoots and undershoots may be tolerable. However, under other circumstances,the nonlinear processes of turbulence diffusion will feed back and amplify theseover/undershoots, and may lead to divergence of a solution. During the past decade,efforts have been made to derive high-resolution and bounded schemes. LODA,SHARP, and SMART all display unbounded behavior, which leads to an increasein the computer storage requirement, especially for three-dimensional flow cal-culations. Therefore, the traditional method for simulating turbulent flows is thehybrid (upwind/central differencing) scheme, and the upwind is used for turbulenceequations such as kinetic energy equation, dissipation rate equation, and Reynoldsstress equations. Since it has a poor track record, one should always be suspiciousof the first-order upwind scheme.



1.2 Numerical Formulation

1.2.1 Governing equations

The conservation equations governing incompressible steady flow problems areexpressed in the following general form:

div[ρ V− grad()] = S (1)

where is any transport variable, V the velocity vector, ρ the density of the fluid, the diffusive coefficient, and S is the source term of variable .

With ξ, η, and ζ representing the general curvilinear coordinates in three-dimensional framework, the transport equation (1) can be expressed as:

1

J

[∂ρU

∂ξ+ ∂ρV

∂η+ ∂ρW

∂ζ

]= 1

J

∂

∂ξ

[

J(q11ξ)

]+ 1

J

∂

∂η

[

J(q22η)

]+ 1

J

∂

∂ζ

[

J(q33ζ)

]+ SCD + S(ξ, η, ζ)

(2)

where U , V , and W are contravariant velocities defined as follows:

U = j11u + j21v+ j31w (3a)

V = j12u + j22v+ j32w (3b)

W = j13u + j23v+ j33w (3c)

J is the Jacobian coefficient, qij and jij (i = 1 − 3 and j = 1 − 3) are the transforma-tion coefficients (refer to the appendix), and SCD is the cross-diffusion term (referto the appendix).

1.2.2 Discretization

The computational domain is uniformly divided into hexahedral control volumes,and the discretization of transport equation (2) is performed in the computationaldomain following the finite-volume method.

Integrating equation (2) over a control volume as shown in Figure 1.1 andapplying the Gauss Divergence Theorem in conjunction with central difference fordiffusion, we have:

Fe − Fw + Fn − Fs + Ft − Fb = SV + SCDV (4)

where Ff represents the total fluxes of across the cell face f (f = e, w, b, s, b, t).



W

z T

w

t

s

S

b

b

Pe

x E

n

Figure 1.1. A typical control volume.

Taking the east face as an example, the total fluxes across it can be written as:

Fe = (ρU)e −(

JJ11

)e(E −P) (5)

In the above equation, the cell face values of can be approximated with differentschemes.

For the first-order upwind scheme, the cell face value is expressed as:

e = P if Ue > 0e = E if Ue < 0

(6)

Substituting equations (5) and (6) into equation (4), we have:

APP =∑

i=E,W,N,S,T,B

Aii + SC (7)

where subscript i denotes neighboring grid points, AP and Ai the coefficients relatingto the convection and diffusion, and Sc is the source term.

1.2.3 Higher-order schemes

The approximation of convection has a decisive influence on the overall accuracy ofthe numerical simulations for a fluid flow. The first-order schemes such as upwind,hybrid, and power-law all introduce the second-order derivatives that then leadto falsely diffusive simulated results. Therefore, the higher-order schemes have



to be used to increase the accuracy of the solution. Generally, with uniform gridspacing, the higher-order interpolation schemes can be written in the followingweighted-average form:

e = P + 1

4[(1 − κ)−

e + (1 + κ)e] if Ue > 0

e = E − 14

[(1 − κ)+e + (1 + κ)e] if Ue < 0 (8)

where −e =P − W,e =E −P,+

e =EE −E and κ is the weighted-average coefficient. In equation (8), the underlined terms represent the fragmentsof the first-order upwind scheme. Therefore, the higher-order schemes can beimplemented in a deferred correction approach proposed by Khosla and Rubin[9]; that is,

n+1f = UP,n+1

f + (HO,nf −UP,n

f ) (9)

where n indicates the iteration level, and UP and HO refer to the upwind andhigher-order schemes, respectively. The convective fluxes calculated by the upwindschemes are combined with the diffusion term to form the main coefficients of thedifference equation, while those resulting from the deferred correction terms arecollected into the source term, say, SDC. Such a treatment leads to a diagonallydominant coefficient matrix and enables a higher-order accuracy to be achieved ata converged stage.

With this method, the deferred correction source term, taking east–westdirection as an example, is calculated by:

SDC = 1

4

U+

e Ue[(1 + κ)e + (1 − κ)−e ] − U−

e Ue[(1 + κ)e + (1 − κ)+e ]

− U+w Uw[(1 + κ)w + (1 − κ)−

w] + U−w Uw[(1 + κ)w + (1 − κ)+

w]

(10)

where U±e is defined as:

U±f = 1 ± sgn(Uf )

2If κ is fixed at a suitable constant value everywhere, several well-known schemescan be formed.

However, the schemes listed inTable 1 all suffer from boundedness problem; thatis, the solutions may display unphysical oscillations in regions of steep gradients,which can be sufficiently serious to lead to numerical instability.

1.2.4 Weighted-average coefficient ensuring boundedness

Based on the variable normalization proposed by Leonard [4], with a three-nodestencil as shown in Figure 1.2, we introduce a normalized variable defined as:

= −U

D −U(11)



Table 1.2. Typical interpolation schemes

Expression for e when u> 0 Leading truncation error term

1/2(3P −W) 3/8x3′′

1/2(E +P) 1/8x2′′

1/8(3E + 6P −W) 1/16x3′′′

1/6(2E + 5P −W) −1/24x2′′

FU

W EPw e

FC

FfFD

Figure 1.2. Three-node stencil.

where the subscripts U and D represent the upstream and downstream loca-tions, respectively. In the normalized form, the higher-order schemes can berewritten as:

f = C + 1

4[(1 + κ)(1 − C) + (1 − κ)C] (12)

See Figure 1.2 for notations of the terms. Solving for κ,

κ = 4f − 4C − 1

1 − 2C(13)

In order to ensure boundedness, the TVD constraints can be used; that is,

f ≤ 1, f ≤ 2C, f ≥ C for 0 < C < 1

f = C for C ≤ 0 or C ≥ 1 (14)

which correspond to the triangle region shown in Figure 1.3.



TVDconstraints

A

1.0

B 1.0

CFc

WACEB

QUICK

~

2F c~F f

~

F c~

Figure 1.3. Diagrammatic representation of the TVD constraint and WACEBscheme.

The Taylor series expansion shows that the first two leading trunca-tion error terms of the interpolation scheme (7) are 1/4(κ− 1/2)x2′′ and1/8(1 − κ)x3′′′. Therefore, the scheme has at least the second-order accuracy.The maximum accuracy (third order) can be achieved if κ is set equal to 1/2. Thus,the scheme can be formed in such a way that κ lies as close as possible to 1/2, whilesatisfying the TVD constraints. Based on this idea, the normalized cell face valuecan be computed by the following expressions:

f =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩C C /∈ [0, 1]

2C C ∈ [0, 0.3)

3/8 (2C + 1) C ∈ [0.3, 5/6]

1 C ∈ (5/6, 1]

(15)

As shown in Figure 1.3, TVD constraints are overly restrictive according toconvection boundedness criterion (CBC). However, the use of a larger multiplyingconstant will not noticeably increase the accuracy. The reasons are that, first, theconstant affects the accuracy only in the range from A to B (see Figure 1.3), andthis range varies at most from 0 to 0.3 (if we use constant 3, A = 0.1666 andB = 0.3). Secondly, even with the smaller constant, the accuracy of the scheme isstill second order. Therefore, the present WACEB (weighted-average coefficientensuring boundedness) scheme employs normalized variable formulation (15) tocalculate the weighted-average coefficient to preserve boundedness.



1.00

0.50

0.00κ

−0.5

−1.0−2.5 −0.0 2.50

Φc∼

Figure 1.4. The variation in weighted-average coefficient with normalized variable.

From equations (7) and (8), the weighted-average coefficient can be given by:

κ =

⎧⎪⎪⎪⎨⎪⎪⎪⎩1/(1 − 2C) C /∈ [0, 1](4C − 1)/1 − 2C C ∈ [0, 0.3)(3 − 4C)/(1 − 2C) C ∈ [0.3, 5/6]1/2 C ∈ (5/6, 1]

(16)

The variation in κ with C is shown in Figure 1.4. It is easy to see that the presentWACEB scheme satisfies convective stability condition [2]. It is necessary to men-tion that the above algorithm is formulated on the assumption of the constant gridspacing. For nonuniform grids, the weighted-average coefficient will also be thefunction of the grid spacing aspect ratio.

1.3 Test Problem and Results

The governing transport equations are solved by using the nonstaggered finite-volume method. A special interpolation procedure developed by Rhie and Chow[10] is used to prevent pressure oscillations due to nonstaggered grid arrangement.Pressure and velocity coupling is achieved through the SIMPLE algorithm [8].

It is necessary to mention that QUICK and WACEB schemes all need to employtwo upstream nodes for each cell face, which mandates one to involve a valueoutside the solution domain for a near-boundary control volume. Therefore, theupwind scheme is used for all the control volume adjacent to boundaries.

1.3.1 Pure convection of a box-shaped step profile

The flow configuration shown in Figure 1.5 constitutes a test problem for examiningthe performance of numerical approximation to convection because of the extremely



1.00.15

0.15Φ = 0

v

u

1.0

0.15

0.15

Φ = 1

Φ = 0

V

Figure 1.5. Pure convection of a box-shaped step by a uniform velocity field.

sharp gradient in a scalar. This is a linear problem in which the velocity field isprescribed. The calculations are performed with two different uniform meshes,29 × 29 and 59 × 59.

Comparisons of the numerical solutions obtained with the upwind, QUICK,and WACEB schemes are presented in Figure 1.6(a) and (b). It can be seen thatthe upwind scheme results in a quite falsely diffusive profile for the scalar evenwith the finer mesh. Although the QUICK scheme reduces such a false diffusion,it produces significant overshoots and undershoots. Unlikely, the WACEB predictsa fairly good steep gradient without introducing any overshoots or undershoots.Therefore, we conclude that the WACEB scheme resolves the boundedness problemwhile reserving a higher-order accuracy.

1.3.2 Sudden expansion of an oblique velocity field in a cavity

The geometry under consideration is depicted in Figure 1.7. The flow is assumed tobe steady and laminar. At the inlet, U-velocity and V-velocity are given a constantvalue of Uref . The boundary conditions at the outlet are ∂U/∂x = 0 and ∂V/∂x = 0.The calculations are performed on the uniform meshes (59 × 59). Figure 1.8 showsthe comparison of U-velocity along the vertical central lines of the cavity for theReynolds number 400. It is noticed that the upwind scheme cannot predict thesecondary recirculation region well, which should appear near the upper side ofthe cavity and smears out the steep gradients of the velocity profile near the main-stream. We observe that both theWACEB and QUICK schemes distinctively predictthis secondary recirculating region. Furthermore, it is noteworthy to observe thatboth produce very similar results. The streamline patterns predicted with the threeschemes are all shown in Figure 1.9. It is clearly seen, again, that the upwind



1.250

1.000

0.750

0.500

0.250

0.000

0.000 0.250 0.500 0.750 1.000−0.250

Φ

Y29 × 29

1.250

(b)(a)

1.000

0.750

0.500

0.250

0.000

0.000 0.250 0.500 0.750 1.000−0.250

Φ

Y59 × 59

ExactQuickWacedUpwind

ExactQuickWacedUpwind

Figure 1.6. Scalar profiles along the center line.

L

y

Lx0.15

L

0.15

L

Figure 1.7. Geometry of a cavity.

scheme predicts a much smaller vortex on the upper left side of the cavity andmuch wider mainstream region than the QUICK and WACEB schemes. The com-putations were further extended to a higher Reynolds number up to 1,000. At thisReynolds number, the QUICK scheme produces a “wiggle solution.” Figure 1.10shows streamline patterns predicted with the WACEB and upwind schemes. Thesetwo schemes give very different flow patterns; with the increase in the Reynoldsnumber, the convection is enhanced and diffusion is suppressed and then the “deadwater regions” should have less effect on the mainstream region. The results with



0.250

0.500

QUICK

WACEB

UPwind

0.750U

/Ure

f

0.000

0.000 0.250 0.500

y /L

0.750 1.000−0.25

Figure 1.8. U-velocity profile along the vertical center line of the domain(Re = 400).

1.00

0.75

0.50

0.25

0.000.00 0.25 0.50

x/L0.75 1.00

y/L

(a) (b)1.00

0.75

0.50

0.25

0.000.00 0.25 0.50

x/L0.75 1.00

y/L

(c)1.00

0.75

0.50

0.25

0.000.00 0.25 0.50

x/L0.75 1.00

y/L

Figure 1.9. Streamlines for sudden expansion of an oblique velocity field(Re = 400): (a) QUICK; (b) WACEB; (c) upwind.

the WACEB scheme clearly show this trend. It is also noted that the WACEBscheme produces two additional vortices at the two corners of the cavity. However,the upwind scheme predicts only a very small additional vortex at the lower rightcorner and fails to capture the additional vortex at the upper left corner.



1.00

0.75

0.50

0.25

0.000.00 0.25 0.50

x/L0.75 1.00

1.00

0.75

0.50

0.25

0.000.00 0.25 0.50

x/L0.75 1.00

y/L

y/L

(a) (b)

Figure 1.10. Streamlines for sudden expansion of an oblique velocity field(Re = 1,000): (a) WACEB; (b) upwind.

0.1H

6H 30H

H

S =

0.7

5H

Figure 1.11. Geometry of flow over a fence.

From the above discussions, it is concluded that the solution with the WACEBscheme is comparable to that with the QUICK scheme. Even under highly con-vective conditions in which the unbounded QUICK scheme may produce “wigglesolutions,” the bounded WACEB scheme still produces a reasonable solution.

1.3.3 Two-dimensional laminar flow over a fence

A two-dimensional laminar flow over a fence (see Figure 1.11) with the Reynoldsnumber based on the height of the fence, the mean axial velocity of 82.5, and theblockage ratio (s/H ) of 0.75 is a benchmark case study. The boundary conditionsat the inlet are prescribed as a parabolic profile for the axial velocity U and zerofor the cross-flow velocity V . At the outlet, the boundary conditions are given as∂U/∂x = 0 and ∂V/∂x = 0. The present study shows that the grid-independenceresults can be achieved with 150 × 78 uniform meshes for all the schemes.



1.0

0.8

0.6

0.4

y/H

y/H

0.2

0.0−6.8 −5.3 0 1.2 2.0

x/s

1.0

0.8

0.6

0.4

0.2

0.04.0 5.0 6.0 8.8

x/s

(a)

(b)

1.5

Figure 1.12. Comparison between prediction and measurement for flow over thefence (Re = 82.5) (square symbol, experimental data; solid line,WACEB; dashed line, QUICK; dash–dot line, upwind)

Figure 1.12 presents the axial velocity profiles at different locations (x/s) mea-sured [11] and calculated with the QUICK, WACEB, and upwind schemes. We canobserve that when x/s is less than 2, the results with the three schemes are nearlyidentical and are in good agreement with experimental data. However, when x/s islarger than 2, where the second separated flow on the top wall appears, the upwindscheme predicts very poor results and the QUICK and WACEB schemes give verysatisfactory results in comparison with the experimental data [11]. These resultsverify the conclusion drawn from previous section.

1.4 Conclusions

By using normalized variable formulation and TVD constraints, the WACEB ofthe solution is determined and then a bounded scheme is presented in this chapter.This new scheme is tested for four different flow applications including a linearconvection transport of a scalar, a sudden expansion of an oblique flow field, and a



laminar flow over a fence. The numerical tests show that the new WACEB schemeretains the ability of the QUICK to reduce the numerical diffusion without intro-ducing any overshoots or undershoots. The scheme is very easy to implement,stable, and free of convergence oscillation and does not need to incorporate anyunder-relaxation treatment for weighted-average coefficient calculation.

Appendix

The cross-diffusion source term in equation (2) is defined as follows:

SCD = 1

J

∂

∂ξ

(

J(q21η + q31ζ)

)+ 1

J

∂

∂η

(

J(q12ξ + q32ζ)

)+ 1

J

∂

∂ζ

(

J(q13ξ + q23η)

)The transformation coefficients are defined as follows:

j11 = ∂y

∂η

∂z

∂ζ− ∂y

∂ζ

∂z

∂η, j12 = ∂y

∂ζ

∂z

∂ξ− ∂y

∂ξ

∂z

∂ζ, j13 = ∂y

∂ξ

∂z

∂η− ∂y

∂η

∂z

∂ξ,

j21 = ∂x

∂ζ

∂z

∂η− ∂x

∂η

∂z

∂ζ, j22 = ∂x

∂ξ

∂z

∂ζ− ∂x

∂ζ

∂z

∂ξ, j23 = ∂x

∂η

∂z

∂ξ− ∂x

∂ξ

∂z

∂η,

j31 = ∂x

∂η

∂y

∂ζ− ∂x

∂ζ

∂y

∂η, j32 = ∂x

∂ζ

∂y

∂ξ− ∂x

∂ξ

∂y

∂ζ, j33 = ∂x

∂ξ

∂y

∂η− ∂x

∂η

∂y

∂ξ

and

qij =3∑

k=1

jki jkj (i = 1, 3, j = 1, 3)

Nomenclature

A coefficients in equation (7)F total fluxes across the cell facesH height of channelJ determinant of Jacobianjij , qij (i = 1, 3 and j = 1, 3) transformation factorL length of cavityRe Reynolds numberS, SC , SCD, SDC source terms hight of fenceU , V , W contravariant velocity componentsUm mean velocity in the channelu, v, w Cartesian velocity componentsx, y, z Cartesian coordinates



Greek Symbols

diffusion coefficientκ weighted-average coefficient dependent variableξ, η, ζ generalized curvilinear coordinates

Superscript

HO term associated with higher-order schemeUP term associated with upwind schemen iteration level∼ normalized value

Subscripts

f (=e, w, n, s, t, b) value at the cell facesF (=E,W, N, S,T, B) value at the nodes

References

[1] Price, H. S. Varga, R. S., and Warren, J. E. Application of oscillation matrices todiffusion-correction equations, J. Math. Phys., 45, pp. 301–311, 1966.

[2] Leonard, B. P.A stable and accurate convective modelling procedure based on quadraticupstream interpolation, Comput. Methods Appl. Mech. Eng., 19, pp. 59–98, 1979.

[3] Zhu, J., and Leschziner, M. A. A local oscillation-damping algorithm for higher-orderconvection schemes, Comput. Methods Appl. Mech. Eng., 67, pp. 355–366, 1988.

[4] Leonard, B. P. Simple high-accuracy resolution program for convective modelling ofdiscontinuities, Int. J. Numer. Methods Fluids, 8, pp. 1291–1318, 1988.

[5] Gaskell, P. H., and Lau, A. K. C. Curvature-compensated convective transport:SMART, a new boundedness preserving transport algorithm, Int. J. Numer. MethodsFluids, 8, pp. 617–641, 1988.

[6] Zhu, J. On the higher-order bounded discretization schemes for finite volume compu-tations of incompressible flows, Comput. Methods Appl. Mech. Eng., 98, pp. 345–360,1992.

[7] Zhu, J. A low-diffusive and oscillation-free convection scheme, Comm. Appl. Numer.Methods, 7, pp. 225–232, 1991.

[8] Patankar, S. V. Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York1980.

[9] Khosla, P. K., and Rubin, S. G. A diagonally dominant second-order accurate implicitscheme, Comput. Fluids, 2, pp. 207–209, 1974.

[10] Rhie, C. M., and Chow, W. L. A numerical study of the turbulent flow past an isolatedairfoil with trailing edge separation, AIAA J., 21, pp. 1525–1532, 1983.

[11] Carvalho, M. G. Durst, F., and Pereira, J. C. Predictions and measurements of laminarflow over two-dimensional obstacles, Appl. Math. Model., 11, pp. 23–34, 1987.


2 Higher-order numerical schemes forheat, mass, and momentumtransfer in fluid flow

Mohsen M. M. Abou-Ellail, Yuan Li, and Timothy W. TongThe George Washington University, Washington, DC, USA

Abstract

A novel numerical procedure for heat, mass, and momentum transfer in fluidflow is presented in this chapter. The new scheme is based on a non-upwind,interconnected, multigrid, overlapping (NIMO) finite-difference algorithm. Intwo-dimensional (2D) flows, the NIMO algorithm solves finite-difference equa-tions for each dependent variable on four overlapping grids. The finite-differenceequations are formulated using the control-volume approach, such that no interpo-lations are needed for computing the convective fluxes. For a particular dependentvariable, four fields of values are produced. The NIMO numerical procedure istested against the exact solution of two test problems. The first test problemis an oblique laminar 2D flow with a double-step abrupt change in a passivescalar variable for infinite Peclet number. The second test problem is a rotatingradial flow in an annular sector with a single-step abrupt change in a passivescalar variable for infinite Peclet number. The NIMO scheme produces essen-tially the exact solution using different uniform and nonuniform square andrectangular grids for 45- and 30-degree angles of inclination. All other schemesare unable to capture the exact solution, especially for the rectangular andnonuniform grids. The NIMO scheme is also successful in predicting the exactsolution for the rotating radial flow, using a uniform cylindrical-polar coordi-nate grid. The new higher-order scheme has also been tested against laminarand turbulent flow in pipes as well as recalculating flow behind a fence. Thetwo laminar test problems are predicted with a very high accuracy. The turbu-lent flow in pipes is well predicted when the low Reynolds number k-e modelis used.

Keywords: Discretization scheme, Non-upwind, Interconnected grids, Convectivefluxes



2.1 Introduction

Finite-difference numerical simulations have suffered from false diffusion, whichis synonymously referred to as numerical diffusion. This deficiency and other errorsin computational fluid dynamics (CFD) are an inevitable outcome of the differentinterpolation schemes used for the convective terms. The interpolation schemes forthe convective terms are classified as one-point schemes such as first-order upwind,two-point schemes such as central differencing (CD), and hybrid scheme, whichis a combination of CD and upwind differencing [1–4]. Higher-order schemes,such as three-point second-order (CUI) [5], third-order quadratic interpolation forconvective kinetics (QUICK [6] and QUICK-2D [7]), four-point third-order inter-polation (FPTOI), and four-point fourth-order interpolation (FPFOI) [8], offer aroute to improving the accuracy of the computations. The QUICK-2D scheme isan extension of the QUICK algorithm to enhance its stability in elliptic fluid flowproblems [7]. It utilizes a six-point quadratic interpolation surface that favors thelocally upstream points [7]. All of the above-mentioned schemes were unable topredict the exact profile along the y-axis on the mid-plane of the first test problemof the 45-degree oblique flow with infinite Peclet number [8–10]. These schemesproduced uncertainties ranging from false diffusion and numerical instabilities toovershooting and undershooting [8]. Song et al. [9,10] introduced a higher-orderbounded discretization algorithm (weighted-average coefficient ensuring bounded-ness, WACEB) to overcome overshooting and undershooting encountered in theirprevious FPTOI and FPFOI schemes [8]. They were able to remove the overshootingand undershooting in their numerical results of the 45-degree oblique flow. Eventhe higher-order schemes, mentioned above, were unable to predict the infinitelysteep gradient of the scalar variable φ as it abruptly changes from 0 to 1 in the testproblem [8–10]. Raithby [11] presented a skew differencing scheme that utilizesthe upstream values prevailing along the local velocity vectors at the four faces ofthe 2D control volume surrounding each grid node. This skew differencing schemewould capture most of the details of the oblique flow if the grid is aligned alongthe local velocity vectors. Raithby obtained accurate results for the problem of stepchange of a passive scalar using a square uniform grid with dimensions 11 × 11when the flow is inclined by a 45-degree angle [11]. In this case one of the diagonalsof the control volume surrounding each node is aligned along the uniform veloc-ity field, while the other diagonal is perpendicular to flow direction. Verma andEswaran [12] used overlapping control volumes to discretize the physical solutiondomain. They obtained finite-difference equations that favor the upwind nodes fromwhich the incoming flow emanates [12]. They tested their scheme using a 11 × 11grid for the step change in passive scalar in an oblique flow problem. They wereunable to capture the exact solution with a square grid for the 45-degree flow withstep change in the passive scalar [12]. Interpolation for the convective transport iscommon to all of the above schemes, causing a varying degree of errors. More-over, most of the above schemes involve upwind differencing, either explicitly orimplicitly. However, the diffusive terms of the fluid flow governing equations aremuch easier and are more accurately modeled in most numerical schemes.


Higher-order numerical schemes for heat, mass, momentum transfer 21

φi, j+2

φi, j +1 φi +1, j+1

φi−2, j φi+2, jφi−1, j

φi−1, j −1 φi+1, j −1

φi+1, j

φi, j −2

φi, j −1

φi, j

φi−1, j+1

Figure 2.1. Control volume of single-grid schemes.

2.2 Single-Grid Schemes

Most of the published numerical procedures use a single grid. The main governingequations can be cast in one general form, namely,

∂(ρukφ)

∂xk− ∂

∂xk

(φ∂φ

∂xk

)= Sφ (1)

where uk is the fluid velocity along coordinate direction xk , while φ stands forany dependent variable such as mass fraction or dimensionless temperature. Asexplained by Abou-Ellail et al. [1], equation (1) can be formally integrated, overthe control volume shown in Figure 2.1, to produce the following finite-differenceequation:

(di+1, j + di−1, j + di, j+1 + di, j−1 − Sp)φi, j

= di+1, jφi+1, j + di−1, jφi−1, j + di, j+1φi, j+1 + di, j−1φi, j−1

+ ci−1, jφi−1/2,j − ci+1, jφi+1/2,j + ci, j−1φi, j−1/2 − ci, j+1φi, j+1/2 + Su

(2)

where di+1, j and ci+1, j are diffusion and convection coefficients to be computed atthe center of the east face of the control volume, midway between the central node(i) and the east node (i+1). They are given as:

di+1, j =(

Aφδx

)at i+1/2,j

(3)

ci+1, j = (ρuA)ati+1/2, j (4)

where ρ is the density, u the velocity along the x-axis, A the east face surface area,δx the distance between nodes i and (i + 1), φ the diffusion coefficient of φ, and Su



and Sp are the coefficients of the integrated source term conveniently expressed as alinear expression. Equations similar to (3) and (4) apply to the west (i − 1, j), south(i, j − 1), and north (i, j + 1) nodes. Unlike the convective terms, the diffusion termsrequire no interpolation for intermediate values of φ. Convective terms, involvingCV face values such as φi+1/2, j , require interpolation between the neighboring gridnodes. The single-grid finite-difference equation can be written, for a central nodalpoint P and neighboring east–west–north–south nodes (E, W, N, S), as:

(aE + aW + aN + aS − Sp)φp = aEφE + aWφW + aNφN + aSφS + Su (5)

The hybrid scheme defines the above finite-difference coefficients as follows:

aE = max[

di+1, j ,

∣∣∣∣12ci+1, j

∣∣∣∣] − 12

ci+1, j (6)

aW = max[

di−1, j ,

∣∣∣∣12ci−1, j

∣∣∣∣] − 12

ci−1, j (7)

In the above equations, max [. . . . ,. . . .] represents the maximum value of thetwo values inside the bracket. Coefficients aN and aS have similar expressions inthe hybrid scheme. The pure upwind differencing scheme has slightly differentexpressions for the finite-difference coefficients.

2.3 New Numerical Simulation Strategy

As mentioned above, most of the existing schemes, whether upwind or higherorder, have a certain degree of false diffusion and/or over- and undershooting. Allthe above-mentioned schemes cannot produce an exact numerical solution to thefirst test problem of a 45-degree oblique flow with step changes in a passive scalarfor square grids [8–10] and particularly for rectangular and nonuniform grids [11].The present simulation strategy is based on removing all the ambiguity of inter-polating for the CV face values of the scalar variable φ. This is done simply bysuperimposing four grids on the 2D solution domain. These grids are arrangedin such a way that each grid uses the remaining grids to obtain directly, withoutany interpolations, the CV face values of the scalar variable used in computingthe convective terms. The present new scheme essentially replaces the interpola-tion process, by finite-difference equations for the CV face values of φ. Therefore,the non-upwind, interconnected, multigrid, overlapping (NIMO) scheme elimi-nates most of the interpolation-based false diffusion that creeps into the numericalresults. In addition to handling the scalar variables, the NIMO system can store thevelocity components, pressure and its correction, on same space locations. In thiscase, the velocity components on one grid will still be located between the pres-sures, and their corrections, on the neighboring grids as preferred by the SIMPLEalgorithm explained by Abou-Ellail et al. [1] and Patankar [2]. The NIMO inter-connected grids share some features with the well-known staggered-grid method



[1,2]. Both methods have displaced grids, relative to a defined main grid. How-ever, the NIMO scheme uses four overlapping grids for each dependent variable,while the staggered-grid method requires only one grid per dependent variable, asexplained byAbou-Ellail et al. [1] and Patankar [2]. While the NIMO scheme needsno interpolations for all dependent variables, the staggered-grid method removesthe need for interpolations only for the velocity components but not for the scalarvariables [1,2].

2.4 Novel Multigrid Numerical Procedure

The new NIMO system is shown in Figure 2.2. The main grid defines the nodeswhere φi, j is located in space. Three other grids are shifted in space where φx

i, j , φyi, j ,

and φxyi, j are located midway between the main-grid nodes, as depicted in Figure 2.2.

The superscripts x and y indicate shifting of grids midway with respect to the main-grid nodes. Moreover, the superscript xy indicates that the grid is shifted diagonallysuch that the shifted nodes occupy the center node between the neighboring fournodes of the main grid. The four-node arrowhead clusters shown in Figure 2.2 areused to indicate the common indices (e.g., i, j) affiliated with φ,φx,φy, and φxy.The spatial locations of φ,φx,φy, and φxy affiliated with cluster (i, j), in the solutiondomain, are (xi, yj), (xi +xi/2, yj), (xi, yj +yi/2), and (xi +xi/2, yj +yi/2),respectively. This cluster technique simplifies greatly the finite-difference equationsof NIMO. It also simplifies the computer coding of the system of equations of theNIMO scheme. The NIMO control volumes CV, CVx, CVy, and CVxy are depicted

φi,j+1 φi +1,j+1φi−1,j+1 φx φx

φyφy

φy

φi,j−1 φi+1,j−1

φy φxy

φx

φi −1,j

φi −1,j−1

φi +1,j

φy

φi,j

φxyφxy

φx φx

i,j

i,j

i,j−1 i+1,j−1

i,j−1

i +1,j

i,j+1

φxy

φx

φyi −1,j−1 i,j−1

i −1,ji −1,j

i−1,j+1

i −1,j

i −1,j−1

i −1,j−1

i,j

Figure 2.2. NIMO coordinate system, defining arrowhead clusters, of nodes withsame indices, but differing in their spatial locations, in main, x, y, andxy grids.



φxy

×

×

φxφy

φxy

φx

φx

φx

φxφi,j

φi,j+1 i,j+1

φy

φi ,j−1

φi −1,j i −1,j

i,j+1

φx φi +1,j

φy

φi,j φy

××

×

×

φxy

φxy

φxyφy

φy

φy

CVxyCVy

CVxCV

×

φyi,j+1

φyφxy φy

φy

φy

φi,j

φy

i−1,ji−1,j φxy

i−1,ji,j φy

i,ji,j+1

i,j+1

i,j+1

φxyi,j+1

i,j

i+1,j

i,j+1

i,j−1

φxy

i,j i+1,j i+1,j

i −1,j i,j

i,j

i+1,j i+1,j

i ,j−1

i ,j−1

Figure 2.3. NIMO control volumes (CV, CVx, CVy, and CVxy) surrounding thenodes of the main, x, y, and xy grids.

in Figure 2.3. The main-grid typical control volume CV is formed by the fourplanes bisecting the distances between the neighboring nodes and the central node(i, j). Control volumes CVx, CVy, and CVxy enclose nodal variables φ x

i, j, φyi, j ,

and φxyi, j . Along the x- or y-coordinate, the faces of these control volumes pass by

the nearest neighboring nodes belonging to any of the four grids, as depicted inFigure 2.3. The acting nodal and face values used for convective fluxes of the scalarvariable φ pertaining to each control volume surrounding each node are shown inthe Figure. It should be mentioned here that the control-volume approach adoptedhere is similar to the finite- volume method [2]. However, in the finite-volumemethod, the solution domain is discretized into node-centered finite volumes [2].With this arrangement, the computations of the convective fluxes pertaining toeach control volume can be computed without the need for interpolation, evenwhen using nonuniform grids. First, the main control volume CV is considered.



Equation (1) is formally integrated over the main control volume surrounding atypical node (i, j) where volume integrals are replaced by their surface integralcounterparts performed over the four faces of the CV shown in Figure 2.3. Theresulting finite-difference equation of the main grid can be written as follows:

(di+1, j + di−1, j + di, j+1 + di, j−1 − Spi, j )φi, j


+ ci−1, jφxi−1, j − ci+1, jφ

xi, j + ci, j−1φ

yi, j−1 − ci, j+1φ

yi, j + Sui, j (8)

The finite-difference mass continuity equation of the main-grid nodes can bewritten as follows:

ci+1, j − ci−1, j + ci, j+1 − ci, j−1 = 0 (9)

Equation (9) indicates, as it should, that the sum of the incoming mass fluxes isequal to the sum of the outgoing mass fluxes. Equations similar to (8) and (9) existfor control volumes CVx, CVy, and CVxy of the other three grids. The convectiveand diffusive fluxes (e.g., di+1, j and ci+1, j) are still given by equations (3) and (4).

Equation (8), together with similar equations for CVx, CVy, and CVxy, rep-resents a closed set of finite-difference equations for φ,φx,φy, and φxy. Althoughthey represent the same scalar variable (φ), they differ in their physical locationsin space. However, the solution of these interconnected nonlinear equations is noteasy, at least with the traditional methods, for example, the tri-diagonal matrix algo-rithm (TDMA). Even for a passive scalar, equation (8) has extra terms that mustbe included as source terms. In this case, the source terms represent passive scalarconvective fluxes from the x- and y-grids to the main grid. Therefore, equation (9)is multiplied by φi, j and is used to modify equation (8). This modification is maneu-vered such that the incoming fluxes of φ are added to the left-hand side, while theoutgoing fluxes of φ are attached to the right-hand side of equation (8). This modi-fication helps stabilize the solutions obtained using TDMA as part of a line-by-linealternating-direction algorithm (ADA). Since TDMA is very economical both instorage and in execution time demands, it has been favored over 2D-matrix-solveralgorithms. Moreover, iterating between the four grids is inevitable as the solutionof each grid is strongly dependent on the solutions of the other grids. The finalNIMO finite-difference equations are as follows:

(di+1, j + di−1, j + di, j+1 + di, j−1 − Spi, j + ci+1, j + ci−1, j + ci, j+1 + ci, j−1)φi, j


+ (ci−1, jφxi−1, j + ci−1, jφi, j) + (ci+1, jφi, j − ci+1, jφ

xi, j) (10)

+ (ci, j−1φyi, j−1 + ci, j−1φi, j) + (ci, j+1φi, j − ci, j+1φ

yi, j) + Sui, j

The terms c and c are convective fluxes written in a general form to allowincoming fluxes to be transferred to the left-hand side while outgoing ones appear



only in the right-hand side of equation (10). The new convective fluxes are definedalong the E–W directions as:

ci−1, j =∣∣∣∣12ci−1, j

∣∣∣∣ + 12

ci−1, j (11)

ci−1, j = ci−1, j − ci−1, j (12)

ci+1, j =∣∣∣∣12ci+1, j

∣∣∣∣ − 12

ci+1, j (13)

ci+1, j = ci+1, j + ci+1, j (14)

Similarly, along the N–S direction, c and c are defined as:

ci, j−1 =∣∣∣∣12ci, j−1

∣∣∣∣ + 12

ci, j−1 (15)

ci, j−1 = ci, j−1 − ci, j−1 (16)

ci, j+1 =∣∣∣∣12ci, j+1

∣∣∣∣ − 1

2ci, j+1 (17)

ci, j+1 = ci, j+1 + ci, j+1 (18)

It can be easily shown that the sum of the four components of c is equal to thesum of the four components of c; this is done by adding equations (12), (14), (16),and (18); that is,

ci−1, j + ci+1, j + ci, j−1 + ci, j+1 = ci−1, j + ci+1, j + ci, j−1 + ci, j+1 (19)+ (ci+1, j − ci−1, j + ci, j+1 − ci, j−1)

The sum of terms in parentheses on the right-hand side of equation (19) is equalto zero, as given by equation (9). Further simplification can be achieved if thediffusion fluxes and the convective fluxes are paired, namely,

(ai+1, j + ai−1, j + ai, j+1 + ai, j−1 − Spi, j )φi, j

= di+1, jφi+1, j + di−1, jφi−1, j + di, j+1φi, j+1 + di, j−1φi, j−1(20)

+ (ci−1, jφxi−1, j + ci−1, jφi, j) + (ci+1, jφi, j − ci+1, jφ

xi, j)

+ (ci, j−1φyi, j−1 + ci, j−1φi, j) + (ci, j+1φi, j − ci, j+1φ

yi, j) + Sui, j

It should be mentioned that equation (20) is still equivalent to equation (10) orequation (8). The coefficient a on the left-hand side of equation (20) is similar to



the upwind differing coefficient, in as much as they both represent the net transportby convection and diffusion. However, the right-hand side has a completely differ-ent structure as it accepts the outgoing fluxes. These outgoing fluxes are reducedto zero in the pure upwind scheme. Equation (20) is found to have better conver-sion characteristics than the final NIMO equation (10). The above coefficients aredefined as:

ai+1, j = di+1, j + ci+1, j (21)

ai−1, j = di−1, j + ci−1, j (22)

ai, j+1 = di, j+1 + ci, j+1 (23)

ai, j−1 = di, j−1 + ci, j−1 (24)

Similarly, the NIMO finite-difference equations for control volumes CVx, CVy,and CVxy can be written as:

(axi+1, j + ax

i−1, j + axi, j+1 + ax

i, j−1 − Sxi, jp )φx

i, j

= dxi+1, jφ

xi+1, j + dx

i−1, jφxi−1, j + dx

i, j+1φxi, j+1 + dx

i, j−1φxi, j−1

(25)+ (cx

i−1, jφi, j + cxi−1, jφ

xi, j) + (cx

i+1, jφxi, j − cx

i+1, jφi+1, j)

+ (cxi, j−1φ

xyi, j−1 + cx

i, j−1φxi, j) + (cx

i, j+1φxi, j − cx

i, j+1φxyi, j) + S

xi, ju

(ayi+1, j + ay

i−1, j + ayi, j+1 + ay

i, j−1 − Syi, jp )φy

i, j

= dyi+1, jφ

yi+1, j + dy

i−1, jφyi−1, j + dy

i, j+1φyi, j+1 + dy

i, j−1φyi, j−1

(26)+ (cy

i−1, jφxyi−1, j + cy

i−1, jφyi, j) + (cy

i+1, jφyi, j − cy

i+1, jφxyi, j)

+ (cyi, j−1φi, j + cy

i, j−1φyi, j) + (cy

i, j+1φyi, j − cy

i, j+1φi, j+1) + Syi, ju

(axyi+1, j + axy

i−1, j + axyi, j+1 + axy

i, j−1 − Sxyi, jp )φxy

i, j

= dxyi+1, jφ

xyi+1, j + dxy

i−1, jφxyi−1, j + dxy

i, j+1φxyi, j+1 + dxy

i, j−1φxyi, j−1

(27)+ (cxy

i−1, jφyi, j + cxy

i−1, jφxyi, j) + (cxy

i+1, jφxyi, j − cxy

i+1, jφyi+1, j)

+ (cxyi, j−1φ

xi, j + cxy

i, j−1φxyi, j) + (cxy

i, j+1φxyi, j − cxy

i, j+1φxi, j+1) + S

xyi, ju

The convective fluxes pertaining to each of the four NIMO grids are shown in theFigure 2.4. The proper locations of the fluxes entering or leaving CV, CVx, CVy,and CVxy are shown clearly in the Figure. Interpretation of the above general NIMOequations is not easy. This interpretation becomes clear after reducing the aboveequation to the test problem, which is shown in Figure 2.5.



φy

ci,j+1

i,j+1

i,j+1

i,j−1

i,j−1

i,j−1

i,j−1

i,j+1

i+1,j i+1,j i+1,ji−1,jφy cy

cy

ci−1,j

ci,j+1

cx

cx

ci,j−1

i,j

CVx

CVxy

ci,j−1

cy

cxy

cxy

φy φxy

φxy

φxycxy φxy φxy

φxy

φyi,j i,j

φy

φi,j+1

φi+1,j φxφx φx φx

φx

φi−1,j φ+j φi+1,j

φi,j−1

CV

CVy

×

××

×

i−1,j i−1,ji−1,j i−1,j

i,j−1

i−1,j i+1,ji−1,j i+1,j

i,j+1

Figure 2.4. NIMO convective fluxes crossing the faces of CV, CVx, CVy, and CVxy

control volumes of the main, x, y, and xy grids.

u = 1/tanθv = 1

(0, 1)

0.15

0.15

(0.0)0.15/tanθ

0.15/tanθ

(1/tanθ, 0)

(1/tanθ, 1)

x

y

u

v v

F =

0

F =

0

F = 0

θ

θ

F = 0

F = 1

Figure 2.5. Computational domain of the test problem of inclined flow at angle θwith double-step change in passive scalar variable φ from 0 to 1.



2.5 The First Test Problem

The first test problem chosen to check the present NIMO scheme is shown inFigure 2.5. This oblique flow configuration was also used by Song et al. [8–10] totest their fourth-order schemes. A similar test problem with a single-step changein φ was utilized by Raithby [11] and by Verma and Eswaran [12]. The limitingcase of the flow configuration of Figure 2.5 with infinite Peclet number has an exactanalytical solution. In this case, with the absence of diffusion, the value ofφ remainsconstant in the vicinity of the diagonal of the solution domain at a value of 1.0,while φ assumes a value of zero elsewhere, as shown in Figure 2.5. The variable φ isa passive scalar that has a vanishing source term. The test problem solution domainis [(1/tanθ) × 1], while the velocity components (u and v) are equal to 1/tanθ and1.0, respectively. The density is also taken as constant and is equal to 1.0. The widthalong the x-coordinate, for y = 0.0 and φ= 1.0, is equal to (0.15/tanθ), while alongthe y-coordinate, at x = 0.0, it is 0.15, as shown in Figure 2.5. All the above valuesare in SI units. The boundary condition is also shown in Figure 2.5, with φ= 1.0 ina districted zone and zero everywhere on the solution domain boundaries. The cellPeclet number at any node (i, j) is defined as:

Pei, j = |ci, j|/di, j (28)

For uniform convection and diffusion fluxes, the cell Peclet number is also uniform;otherwise it is grid-node dependent.

For the test problem of Figure 2.5, the values of source terms and the fluxescoefficients in equations (20), (25)–(27) are:

Spi,j = 0; Sui,j = 0 (29)

ci−1, j = ci−1, j; ci, j−1 = ci, j−1 (30)

ci−1, j = 0; ci, j−1 = 0 (31)

ci+1, j = 0; ci, j+1 = 0 (32)

ci+1, j = ci+1, j; ci, j+1 = ci, j+1 (33)

Similarly, the values of c and c in the other three control volumes, CVx, CVy, andCVxy , can be obtained. The NIMO finite-difference equations for the test problemcan thus be obtained from equations (20), (25)–(27) as follows:

[di+1, j + (di−1, j + ci−1, j) + di, j+1 + (di, j−1 + ci, j−1)]φi, j

= di+1, jφi+1, j + di−1, jφi−1, j + di, j+1φi, j+1 + di, j−1φi, j−1 (34)

+ ci−1, jφxi−1, j + ci+1, j(φi, j − φx

i, j) + ci, j−1φyi, j−1 + ci, j+1(φi, j − φy

i, j)



[dxi+1, j + (dx

i−1, j + cxi−1, j) + dx

i, j+1 + (dxi, j−1 + cx

i, j−1)]φxi, j

= dxi+1, jφ

xi+1, j + dx

i−1, jφxi−1, j + dx

i, j+1φxi, j+1 + dx

i, j−1φxi, j−1 + cx

i−1, jφi, j

+ cxi+1, j(φ

xi, j − φi+1, j) + cx

i, j−1φxyi, j−1 + cx

i, j+1(φxi, j − φxy

i, j) (35)

[dyi+1, j + (dyi−1, j + cyi−1, j ) + dyi, j+1 + (dyi, j−1 + cyi, j−1 )]φyi, j

= dyi+1, jφyi+1, j + dyi−1, jφyi−1, j + dyi, j+1φyi, j+1 + dyi, j−1φyi, j−1 + cyi−1, jφxyi−1, j

+ cyi+1, j (φyi, j − φxyi, j ) + cyi, j−1φi, j + cyi, j+1 (φyi, j − φi, j+1) (36)

[dxyi+1, j + (dxyi−1, j + cxyi−1, j ) + dxyi, j+1 + (dxyi, j−1 + cxyi, j−1 )]φxyi, j

= dxyi+1, jφxyi+1, j + dxyi−1, jφxyi−1, j + dxyi, j+1φxyi, j+1 + dxyi, j−1φxyi, j−1

+ cxyi−1, jφyi, j + cxyi+1, j (φxyi, j − φyi+1, j ) + cxyi, j−1φxi, j + cxyi, j+1 (φxyi, j − φxi, j+1 )

(37)

It should be mentioned here that all the above convective fluxes are positive inthe case of the test problem shown in Figure 2.5. This is because all the velocitycomponents in the solution domain are positive, as depicted in the Figure.

The above equations share one feature with the pure upwind method. Thisshared feature is the left-hand sides of the above equations. However, the right-hand sides of the NIMO equations are essentially different, as they accommodatethe outgoing fluxes that are ignored in the upwind scheme. Each NIMO controlvolume receives incoming convective fluxes from two other grids; for examplethe main CV receives two incoming convective fluxes from the x- and y-grids. Inaddition to the incoming upwind flux effects, the outgoing fluxes have a uniqueperturbation. They leave traces behind in the control volumes, as can be seen fromthe last terms of equations (34)–(37). These traces are the outcome of the differencesbetween the CV center φ and the leaving φ values. Positive and negative traces arepossible. While the incoming fluxes of φ tend to increase the CV central values,the traces may increase or decrease φ at the center of the control volume of thegrid in question. Each grid receives fluxes from neighboring grids as well as tracesresulting from the leaving fluxes of φ to the same neighboring grids, as can beseen from the above set of equations defining the NIMO scheme. The net gainin each control volume is redistributed by the diffusion mechanism, as dictatedby the NIMO finite-difference equations above. The interpretations of the reducedNIMO finite-difference equations, reflecting the test problem flow conditions, applyequally to their general counterparts.

2.6 Numerical Results of the First Test Problem

The solution of the test problem is obtained numerically. The TDMA, line-by-line,ADA is adopted [1]. Equations (34)–(37) are modified at the nodes adjoining the



QUICK

FPTOI

FPTOI

WACEB

NIMO

Exactsolution

0.25 0.5Y

0.75 1−0.25

0

0.25

0.5F

0.75

1

1.25

0

Figure 2.6. Profiles of φ for previous numerical schemes and NIMO at x = 0.5 andPe = ∞ for 59 × 59 square grid and θ= 45 degrees.

boundary to satisfy the conditions imposed there. The value of the cell Peclet num-ber is fixed at a very large number to simulate Pe = ∞ of the limiting case of thetest problem. A value of Pe> 104 is found sufficient enough to essentially sup-press diffusion transport from equations (34)–(37). The NIMO simulation resultsdepicted in Figure 2.6 are obtained for a fixed value of Pe = 106. The NIMO solu-tion procedure computes the diffusive fluxes from the given Peclet number andequation (28). Different grids are used for the 45-degree oblique flow for the fourgrids of the NIMO scheme. These grids are as follows: one square grid (59 × 59),two rectangular grids (59 × 69 and 59 × 53), and one nonuniform grid (59 × 69).However, a uniform 99 × 59 grid is utilized for the 30-degree oblique flow. Thesedifferent meshes are used to explore the behavior of the NIMO scheme under con-ditioned imposed by square, rectangular, and nonuniform grids. Under-relaxationfactors (URFs) in the range of 0.3 to 0.5 for the 45-degree oblique flow and 0.6for the 30-degree oblique flow are used for the finite-difference equations of eachgrid. The above ranges are convenient to control the conversion processes of theinterlinked variables φ, φx, φy, and φxy of the NIMO scheme. The number of iter-ations and the residual errors of each grid are discussed below. Figure 2.6 depictsthe results of Song et al. [8–10] for different single-grid schemes and the presentresults of the hybrid single-grid and NIMO schemes, for θ= 45 degrees. The exactsolution is also depicted in the Figure. The present results of the multigrid system(NIMO) gave a numerical solution that exactly fits the analytical solution of thelimiting case, when Pe = ∞. No overshooting, undershooting, or false diffusioncan be detected from the present results. It can be noticed that the NIMO schemeessentially predicts the proper infinite gradient of φ at y = 0.35 and 0.65, as depictedin Figure 2.6. Moreover, the hybrid scheme, with its dominant false diffusion, gave



Y

F

0 0.2−0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.4 0.6 0.8 1

Exact Solution, Infinate Pelect NumberHybrid, Pe2NIMO, Pe 10NIMO, Pe 50NIMO, Pe 200NIMO, Pe 1000

Figure 2.7. Profiles of φ at x = 0.5 for Pe = 10, 50, 200, and 1,000 and exactsolution for Pe = ∞ for 59 × 59 square grids and θ= 45 degrees.

the least agreement with the analytical solution. The FPFOI, FPTOI, and WACEBschemes are much better than the hybrid scheme. However, FPFOI and FPTOIschemes [8] suffer from overshooting and undershooting and were unable to pre-dict the infinite (dφ/dy) at y = 0.35 and 0.65. Although the WACEB scheme [9, 10]results are properly bounded between 0 and 1, it still could not predict the infinitegradient of φ at y = 0.35 and 0.65, as can be seen from Figure 2.6.

It seemed interesting to see how NIMO works for values of Peclet numberless than infinity. Numerical results of φ are also obtained for the same boundaryconditions and using the 45-degree oblique flow. The NIMO scheme profiles ofφ along the y-axis and at x = 0.5 for Pe = 10, 50, 200, and 1,000 are depicted inFigure 2.7. The hybrid results for Pe> 2 and the analytical solution for Pe = ∞ arereplotted in Figure 2.7 for comparison. It can be seen that as Pe increases, (dφ/dy),at y = 0.35 and 0.65, increases. When Pe = 1,000, (dφ/dy) is so steep it almostreaches infinity.

The four-grid values of the passive scalar variable, that is, φ, φx, φy, and φxy,are plotted in Figure 2.8. The profiles of the scalar variable are plotted at x = 0.5for a value of Peclet number equal to 5. The values of φx and φxy at x = 0.5 used inFigure 2.8 have been obtained by interpolating the converged final solution due tothe shifting of the x- and xy-grids with respect to x = 0.5 plane. The four profilesfall on top of each other, indicating that the fields of φ, φx, φy, and φxy convergetogether to essentially the same values. The accuracy of each grid of NIMO isaccessed by computing the sum, over all nodes, of the absolute residual errors.



1.2

1

0.8

0.6

0.4

0.2

0

0 0.2 0.4y

F

0.6 0.8 1

−0.2

Exact solution, infinite pelect numberMain gridX-gridY-gridXY-grid

Pe = 5

Figure 2.8. Profiles of φ, φx, φy, and φxy at x = 0.5 for Pe = 5 for 59 × 59 squaregrids and θ= 45 degrees.

The residual error at a node of each grid is computed as the imbalance of thepertinent finite-difference equation.

The sums of absolute residual errors of the main grid, x-grid, y-grid, and xy-gridare plotted as a percentage of the incoming total fluxes of φ versus the number ofiterations (n) in Figure 2.9, for Pe = 5 and URF = 0.3. The iterations are stoppedwhen the maximum error of any grid is less than 0.1%. This condition is satisfiedwhen n = 975. The above conversion criterion is also almost satisfied near n = 600.However, the larger number of iterations is favored to ensure complete conversion.The pattern of changes of the error curves is very interesting. The main-grid andxy-grid errors are similar and decrease monotonically. However, the x-grid andy-grid errors are nearly identical, as can be seen from Figure 2.9. They convergefaster at the beginning and then suddenly diverge slightly only to converge to thedesired accuracy at 975 iterations. Part of the faster convergence at the beginningis attributed to the fact that four sweeps per iteration were used for the x-grid andy-grid, while only two sweeps per iteration proved to be sufficient for the maingrid and the xy-grid. It is interesting to notice that the four grids interact in sucha way that the main grid and the xy-grid have no direct link. However, they arelinked indirectly through their interactions with both the x-grid and y-grid. On theother hand, the x-grid and y-grid establish their indirect link by interacting directlywith both the main grid and the xy-grid. The interlinking of the grids explains thepairing of the sum of the absolute residual errors of the main grid and the xy-grid aswell as the pairing of the x-grid and y-grid. It should be mentioned that the initial



10.00

1.00

0.10

0.01

0 200 400 600 800Number of iterations (n)

Sum

of a

bsol

ute

resi

dual

err

ors

(%)

1000 1200

Main grid X-grid y-grid xy-grid

Figure 2.9. Percentage sum of absolute residual errors of the finite-difference equa-tions of main, x-, y-, and xy-grids of NIMO for Pe = 5 for 59 × 59square grids and θ= 45 degrees.

1.2

1

0.8

0.6

0.4

0.2

0

0 0.2 0.4y

F

0.6 0.8 1−0.2

Exact solution, infinite peclet numberHybrid, Pe = 1NIMO, Pe = 1

Figure 2.10. Profiles of φ for NIMO and hybrid schemes at x = 0.5 for Pe = 1 for59 × 59grids and θ= 45 degrees.

error, which is approximately equal to 3.4, arises from the initial guess. The exactanalytical solution of the test problem is used as the initial guess for φ, φx, φy, andφxy for all runs reported in the present work.

NIMO and hybrid profiles of φ are depicted in Figure 2.10, for Pe = 1.0 andx = 0.5. In this case, φ is equally transported by convection and by diffusion.



1.2

1.0

0.8

0.6

0.4

F

0.2

0.01.0E+00 1.0E+01 1.0E+02

Peclet number1.0E+03 1.0E+04

Edge value [(x,y )= (0.5,0.65)]

Central value [(x,y )= (0.5,0.5)]

Figure 2.11. NIMO values of φ at edge [(x, y) = (0.5,0.65)] and center of thesolution domain [(x, y) = (0.5,0.5)] versus Peclet number for 59 × 59square grids and θ= 45 degrees.

Figure 2.10 shows that the hybrid scheme produced skewed profiles of φ alongthe y-axis, while NIMO properly predicts symmetrical profiles. The skewed partof φ is an outcome of the inevitable false diffusion of the hybrid scheme, whichresults from blowing wind along the positive direction of the y-axis. This can alsobe detected from the hybrid scheme profile of φ, for Pe> 2.0, which is shown inFigure 2.7. Similar skewed profiles of φ were also obtained by Song et al. [8–10],for Pe> 2.0. It can be seen from Figure 2.10 that the central value of φ is higher inthe case of NIMO. This is because the hybrid scheme suffers from extranumericaldiffusion, as shown in Figure 2.10.

It is also interesting to determine the minimum Peclet number that producesessentially a flat profile of φ along the y-axis for 0.35< y< 0.65. This is achieved inFigure 2.11 by plotting the value of φ along the edge and center of the diagonal edgeversus Pe. A value of 104 is just sufficient for Pe to produce nearly diffusion-freeresults.

The φ contours that produced Figure 2.11 are depicted in Figure 2.12. Theskewed φ profile of the hybrid scheme along the y coordinate direction is alsoobvious from Figure 2.12a. However, the hybrid scheme contours appear to benearly symmetrical around the diagonal of the solution domain, as this is thetrue flow direction. Moreover, the NIMO results indicate that φ is more con-tained inside the zone bounded by y = 0.35 and y = 0.65, as can be seen fromFigure 2.12b.

More contours are plotted in Figure 2.13. The additional NIMO contours arefor Pe = 10, 50, 200, and 1,000. Figure 2.13a shows hybrid scheme φ contoursfor Pe> 2.0. The hybrid contours show that φ remains close to unity in a very



x

y

0.01

0.250.5

0.50.250.01

0.75

0.99

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(a) (b)x

y

0.010.250.5

0.750.75

0.50.250.01

0.75

0.9

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 2.12. Contour plots of φ for unity Peclet number for 59 × 59 square gridsand θ= 45 degrees. (a) Hybrid, Pe = 1. (b) NIMO, Pe = 1.

small area next to the inlet section where φ assumes a value of 1.0. However,the NIMO contours show that the area where φ remains uncontaminated increasesas the Peclet number increases. Finally at Pe = 1,000, most of the core along thediagonal of the solution domain is filled with unity φ fluid, as can be seen fromFigure 2.13e. Moreover, the widths of the narrow strips, bounded by φ= 0.99and 0.01, that surround the diagonal core decrease as Peclet number increases.These narrow strips become nearly nonexistent as the Peclet number increasesbeyond 1,000. In order to find out how much NIMO is dependent on the boundaryconditions, along the exit sections of the solution domain, a few computer runswere performed. These runs had zero φ gradients at the exit section instead of thefixed-φ-value boundary condition. These runs, although not reported here, showedthat the NIMO scheme is not sensitive to this change in boundary condition forthe flow configuration of Figure 2.5, except for the nodes very near to the exitboundary of the solution domain. For these nodes, insignificant differences on theorder of ±0.1%, between zero-gradient and fixed-value boundary conditions, wereencountered.

In order to check the consistency of the NIMO scheme, a number of meshes wereused to compute the oblique flow of Figure 2.5, for θ= 45 degrees. All the gridshave 59 nodes along the x-coordinate, while 53, 59, and 69 nodes are used alongthe y-coordinate. One of these grids is nonuniform (59 × 69), while the other threeare uniform (59 × 53, 59 × 59, and 59 × 69). The scalar variable profiles, along thex = 0.5 plane, are shown in Figures 2.14 and 2.15, for two nominal values of thePeclet number, namely, 50.0 and 1,000. Only the 59 × 59 square grid has uniformPeclet number. However, the rectangular uniform grids have Peclet number valuesalong the y-axis higher or lower than the nominal value along the x-axis. This isbecause x does not equal y for the rectangular grids, while the fluid physicalproperties are uniform and u = v for the 45-degree flow. Moreover, the nonuniformgrid has Peclet numbers that are space dependent. For a nominal Peclet number of50.0, the profiles of φ at x = 0.5 for the above-mentioned grid dimensions are in



x

y

0.01

0.250.5

0.750.75

0.50.25

0.01

0.99

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

(a) (b)

(c) (d)

x

y

0.010.25

0.5

0.75

0.750.5

0.250.01

0.990.99

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0.010.25

0.50.75

0.990.990.75

0.5

0.25

0.01

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0.010.25

0.50.75

0.990.99

0.75

0.5

0.25

0.01

x0 0.2 0.4 0.6 0.8 1

y

0

0.2

0.4

0.6

0.8

1

x

y

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0.01

0.01

0.99

(e)

Figure 2.13. Contour plots of φ for hybrid scheme (Pe> 2) and NIMO system(59 × 59 square grids, θ= 45 degrees and Pe = 10, 50, 200 and 1,000).(a) Hybrid, Pe> 2. (b) NIMO, Pe = 10. (c) NIMO, Pe = 50. (d) NIMO,Pe = 200. (e) NIMO, Pe = 1,000.

good agreement with each other, as depicted in Figure 2.14. The minor differencesbetween them arise from the nonuniformity of the Peclet number of the rectangularand nonuniform grids, as can be seen from the Figure.

The profiles of φ for different grid geometries and dimensions are depicted inFigure 2.15, for a nominal Peclet number of 1,000 and θ= 45 degrees. The 59 × 69



1.2

1.0

0.8

0.6F

0.4

0.2

0.00.0 0.2 0.4

Y

0.6 0.8 1.0

Exact solution, infinite peclect number59 × 69, Pex = 50, Pey = 42.54

59 × 69, irregular grid, Pemin = 9.62, Pemax = 5059 × 53, Pex = 50, Pey = 55.8859 × 59, Pe = 50

Figure 2.14. Profiles of φ at x = 0.5, for different grid dimensions and geometriesfor Peclet numbers along the x-axis equal to 50 and θ= 45 degrees.

1.2

1.0

1.0

0.8

0.8

0.6

0.6Y

0.4

0.4

0.2

0.00.0 0.2

F

Exact solution, infinite peclet number59 × 69, Pex = 1000, Pey = 850.7559 × 59, Pe = 100059 × 53, Pex = 1000, Pey = 1117.6559 × 69, irregular grid, Pemin = 192.41, Pemax = 1000

Figure 2.15. Profiles ofφ at x = 0.5, for uniform and irregular different grid dimen-sions, Peclet numbers along the x-axis equal to 1,000 and θ= 45degrees.



Exact solution, infinite peclet number

30-degree inclined flow, 99 × 59 grid, pex = 1,730, pey = 1,000

1.2

1.0

0.8

0.6F

0.4

0.2

0.00.0 0.2 0.4

y0.6 0.8 1.0

Figure 2.16. Profiles of φ, at x = 0.5/tanθ, for 99 × 59 grids, Peclet number alongy-axis = 1,000, and θ= 30 degrees.

nonuniform grid is uniform along the x-coordinate and nonuniform along the ydirection. The 59 × 69 grid contracts along the y-coordinate from the outer bound-aries toward the center, where y = 0.5. For this grid, the ratio of the maximumincrement to the minimum increment along the y-coordinate is 4.5. The profilesessentially capture the analytical solution for infinite Peclet number and θ= 45degrees, although they were computed for different grid geometries and dimen-sions. It can be concluded that irrespective of the dimensions or geometry of thecomputational grid, the NIMO scheme is capable of computing the exact solution.However, the skewed upstream-differencing scheme of Raithby [11] can capturethe exact solution only for square grids where the diagonals of the computationalcells are parallel and normal to the flow direction.

Another test is introduced in Figure 2.16 for an angle of inclination of 30degrees. In this case, the oblique flow passes diagonally through a rectangularsolution domain. The distance of the solution domain and the velocity componentalong the x-axis are approximately equal to 1.73. The grid used for the profilesin the Figure has dimensions of 99 × 59. The Peclet numbers along the x- and y-axes are 1,730 and 1,000, respectively. Here also none of the computational celldiagonals is parallel or normal to the flow direction. However, the profile of φ cap-tures the exact analytical solution for infinite Peclet numbers, as can be seen fromFigure 2.16.

2.7 The Second Test Problem

The second test problem is chosen in the cylindrical-polar domain to check theaccuracy and applicability of the NIMO numerical scheme in coordinates with



u 1

u 1

v 1/r

φ 0

θ1 1 rad

θφ 0

φ 1

φ 1

u 1

v 1

r

r1 2

r0 1

v 1/2

Figure 2.17. Cylindrical-polar computational domain of the test problem of rotat-ing radial flow in an annular sector with single-step change in passivescalar variable φ from 0 to 1.

curvature. The configuration is that of a rotating radial flow in an annular sector.The velocity field is prescribed as shown in Figure 2.17. The tangential veloc-ity u is taken as uniform and is equal to unity, while the radial velocity v isinversely proportional to the radial distance r. This velocity configuration satis-fies the mass continuity of the constant-density rotating radial flow in an annularsector.

The annular sector is bounded by r0 = 1.0, r1 = 2.0, and θ= 0.0 and 1.0, alongthe radial and tangential directions, respectively. The passive scalar variable φ isequal to 0.0 and 1.0 at the two inlet sections defined by θ= 0.0 and by r = 1.0. Forinfinite Peclet numbers, the exit sections have values of φ equal to 1.0 and 0.0 forθ= 1.0 and r = 2.0, respectively. For this case the exact solution for the streamlinethat divides the sector into a φ= 0.0 zone and another zone with φ= 1.0 is shownin Figure 2.18. The streamline, which runs from point (1,0) to point (2,1), is shownin the Figure, superimposed on the prescribed velocity vector field. The velocityvectors vary in direction and magnitude, which is a good test to the extent of theaccuracy of the NIMO numerical scheme in conditions where false diffusion is notuncommon. Equations (20), (25), (26), and (27) are applicable to 2D cylindrical-polar coordinates flows. The superscripts x, y, and xy, are replaced by θ, r, and rθ,while the distances between the central node and the neighboring nodes becomerδθ and δr, along the tangential and radial directions, respectively. For this testproblem, a 59 × 59 uniform grid is superimposed on the solution domain of anannular sector. The distances along the tangential (angular) direction, which areinvolved in the NIMO equations, increase in the radial direction, although the griditself is uniform.



1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

00 0.5 1 1.5 2

φ = 1

φ = 0

Figure 2.18. Velocity vector field for rotating radial flow in an annual sector, show-ing the boundary curve between regions of φ= 1 and 0 in rθ plane,for infinite Peclet number.

2.8 Numerical Results of the Second Test Problem

The numerical solution of the second test problem is also obtained with the TDMA,line-by-line, ADA. The value of 104 for the cell Peclet number at point (r,θ) = (1,0)is found sufficient to produce nearly diffusion-free profiles of the passive scalarvariable. The URF is taken as 0.5, which produced converged solutions after 500iterations with an error less than 0.1%. The local cell Peclet number along thetangential (angular) direction increases from 104 to 2 × 104, while along the radialdirection it decreases from 104 to 5 × 1,000. This is an outcome of the variationsalong the radial direction of the radial velocity and the tangential distance.

Figure 2.19 shows the angular profiles of the passive scalar variable φ computednumerically and the corresponding step function exact solution. The exact solutionis obtained for the limiting case of infinite Peclet number. The agreement betweenthe two profiles is excellent. The main differences occur in a very narrow angularzone, where φ changes abruptly from 1.0 to 0.0. However, outside this narrow zone,φ is properly computed as zero on one side and unity on the other side.

Similarly, the radial profiles of the passive scalar are depicted in Figure 2.20.The NIMO results show an excellent agreement with the exact solution of a stepfunction change in the passive scalar variable. The gradient of the passive scalaralong the radial direction (δφ/δr) is higher than that along the tangential direction(δφ/rδθ), as can be seen from Figures 2.19 and 2.20. This is an outcome of thehigher mean value of the Peclet number along the radial direction relative to thecorresponding value along the tangential direction.



0 1

1.2

0.8

0.4

0.00.2 0.4 0.6 0.8

θ (rad)

φExact solutionat r 1.5

Numerical dataat r 1.5

10,000 <Peθ < 20,000

5,000 <Per < 10,000

Figure 2.19. Angular profiles of φ, at r = 1.5, for Peclet numbers ranging from5,000 to 10,000 (angular) and from 10,000 to 20,000 (radial), using59 × 59 cylindrical-polar grid.

r

1.2

0.8

φ

0.4

0.01.0 1.2 1.4 1.6 1.8 2.0

θ 0 .5

Exact solutionat θ 0.5

Numerical dataat θ 0.5

10,000 <Peθ < 20,000

5,000 <Per < 10,000

Figure 2.20. Profiles of φ, at θ= 0.5 rad, for Peclet numbers ranging from 5,000 to10,000 (angular) and from 10,000 to 20,000 (radial), using 59 × 59cylindrical-polar grid.



00

0.5

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1.5 21

f 1

f 0

f 0.999

f 0.001

Figure 2.21. Open contours of the passive scalar variable φ, for a 59 × 59 rθ grid.

Figure 2.21 depicts the open contours ofφ in the rotating radial flow in an annularsector. A thin curved strip, where φ varies from 0.001 to 0.999, is computed usingthe NIMO scheme. Outside this narrow strip the passive scalar variable changesabruptly from 0.0 to 1.0. It is interesting to notice the resemblance between theshape of the narrow strip and the exact solution curve that divides the annularsectors to φ= 0.0 and 1.0 zones (Figure 2.18).

The system of finite-difference equations (equations (20), (25)–(27) can furtherbe simplified if the central node (i, j) is replaced by the subscript P, while theneighboring nodes are denoted by the subscripts E, W, N, and S. The actual positionin the xy space is determined by the superscript on the main variable φ to reflectthe different control volumes, CV, CVx, CVy, and CVxy of the NIMO scheme.Therefore, the NIMO equations are simplified as:

aPφP =∑

n

anφn + SN + Su (38)

axPφ

xP =

∑n

axnφ

xn + Sx

N + Sxu (39)

ayPφ

yP =

∑n

aynφ

yn + Sy

N + Syu (40)

axyP φ

xyP =

∑n

axyn φ

xyn + Sxy

N + Sxyu (41)

where the summation∑n

is over the four neighboring nodes (E, W, N, and S) of the

central node P of each grid of the NIMO scheme. The coefficients aP, axP, ay

P, and



axyP are defined as follows:

aP =∑

n

an − SP (42)

axP =

∑n

axn − Sx

P (43)

ayP =

∑n

ayn − Sy

P (44)

axyP =

∑n

axyn − Sxy

P (45)

In the above equations, SP and Su are the coefficients of the linearized sourceterms of the original governing equations. Moreover, SN , Sx

N , SyN , and Sxy

N are extrasource terms resulting from the NIMO scheme. They are given explicitly as:

SN = (cWφP + cWφxW − cWφW) + (cEφP − cEφ

xP − cEφE)

(46)+ (cSφP + cSφyS − cSφS)+ (cNφP − cNφ

yP − cNφN)

SxN = (cx

WφxP + cx

WφP − cxWφ

xW) + (cx

EφxP − cx

EφE − cxEφ

xE)

(47)+ (cxSφ

xP + cx

SφxyS − cx

SφxS) + (cx

NφxP − cx

Nφxyp − cx

NφxN)

SyN = (cy

WφyP + cy

WφxyW − cy

WφyW) + (cy

EφyP − cy

EφxyP − cy

EφyE)

(48)+ (cySφ

yP + cy

SφS − cySφ

yS) + (cy

NφyP − cy

NφN − cyNφ

yN)

SxyN = (cxy

WφxyP + cxy

WφyP − cxy

WφxyW) + (cxy

E φxyP − cxy

E φyE − cxy

E φxyE )

(49)+ (cxyS φ

xyP + cy

SφxP − cxy

S φxyS ) + (cxy

N φxyP − cxy

N φxN − cxy

N φxyN )

In the above equations the finite-difference coefficients aE, aW, aN, and aS arestill given, respectively, by equations (21)–(24). It should be mentioned here that theexact location of a particular grid node is determined by both the subscript and thesuperscript attached to the variable in question. For example, φx

P is the dependentvariable at the central node P of the control volume CVx reflecting the shifting ofthe x-grid with respect to the main grid, as illustrated in Figure 2.3.

2.9 Application of NIMO Scheme to Laminar Flow Problems

The NIMO scheme is now applied to steady flow problems. For constant densityand viscosity laminar flow problems, the governing equations for momentum and



mass are written in tensor form as follows:

∂(ρuiuj)

∂xi− ∂

∂xi

(µ∂uj

∂xi

)= − ∂p

∂xj(50)

∂(ρui)

∂xi= 0 (51)

where µ is the laminar viscosity, uj is the velocity component along the xj coor-dinate, and p is the fluid pressure. For 2D flow uj = (u1, u2) while xj = (x1, x2).On the other hand, if x − y plane is used then uj = (u, v) and xj = (x, y). In thiscase, invoking equations (38)–(41), the finite-difference equations for u and v aregiven as:

auPuP =∑

aunun + AuE(pxP − px

E) + SuN (52)

avPvP =∑

avnvn + AvS(p yS − p y

S) + SvN (53)

where subscripts u and v denote coefficients pertaining to the velocity componentsu and v. Also, Au and Av are areas of main-grid control-volume CV surfacesnormal to u and v, respectively. Similarly, for CVx, CVy, and CVxy the momentumfinite-difference equations are as follows:

axuPux

P =∑

axunux

n + AxuE( pP − pE) + Sx

uN (54)

axvPv

xP =

∑axvnv

xn + Ax

vS( pxyS − pxy

P ) + SxvN (55)

auuPuy

P =∑

ayunuy

n + AyvW( pxy

W − pxyP ) + Sy

uN (56)

ayvPv

yP =

∑ayvnv

yn + Ay

vN( pP − pN) + SyvN (57)

axyuPuxy

P =∑

axyunuxy

n + AxyuE( py

P − pyE) + Sxy

uN (58)

axyvPv

xyP =

∑axyvnv

xyn + Axy

vN( pxP − px

N) + SxyvN (59)

Equations (52)–(59) define the eight velocity components of the four grids ofthe NIMO scheme. Additional equations for p, px, py, and pxy are needed to closethe above equations. This is done by writing the pressure correction equationsfor the NIMO scheme control volume CV, CVx, CVy, and CVxy. These pres-sure correction equations are the counterparts of the mass continuity equations forCV, CVx, CVy, and CVxy control volumes.

The pressure correction equation of CV can be obtained from the correctedvelocity components ux

P , uxW, vy

P , and vyS , which are used in the mass continuity

finite-difference equation of control volume CV. The corrected velocity componentsare given as:

uxP = ux∗

P + DuxE(p′

P − p′E) (60)



uxW = ux∗

W + DuxW(p′

W − p′P) (61)

vyP = vy∗

p + Dvyn(p′

P − p′N ) (62)

vyS = vy∗

S + DvyS(p′

S − p′P) (63)

where uxP , ux

W, vyP, and vy

S are the corrected velocity components pertaining to CV.Corresponding to the above corrected velocity components, the pressure at eachgrid node is corrected to (p∗ + p′). Moreover, the asterisk superscript indicatesmomentum-based velocity components and pressure. In equations (60)–(63), thecoefficients Dux

E and DvyN are defined as:

DuxE =

(∂u

∂p

)E

= AxuE

axuP

(64)

DvyN =

(∂v

∂p

)N

= AyvN

ayuP

(65)

Similar expressions exist for DuxW and Dvy

S. Substituting equations (60)–(63) inthe mass continuity equation of CV, the corresponding pressure correction equationcan be obtained as follows:

aPp′P =

∑anp′

n − δmP (66)

where p′ is the pressure correction of the main grid, while δmp is the excess massleaving control volume CV per unit time. It is defined explicitly as:

δmP = ρEAxuEux

P − ρWAxuWux

W + ρNAyvNv

yP − ρSAy

vSvyS (67)

The finite-difference coefficient an stands for aE, aW, aN, and aS, and can bedefined as:

aE = ρEAxuEDux

E (68)

aN = ρNAyvNDvy

N (69)

Similar expressions exist for aW and aS. The remaining pressure correctionequations for CVx, CVy, and CVxy are similarly obtained as:

axPp′x

P =∑

axn p′x

n − δxmP (70)

ayPp′y

P =∑

ayn p′y

n − δymP (71)

axyP p′xy

P =∑

axyn p′xy

n − δxymP (72)



Equations (52)–(57), (66), and (70)–(72) represent a complete set of finite-difference equations for u, v, ux, vx, uy, vy, uxy, vxy, p′, p′x, p′y, and p′xy of the NIMOscheme.The above finite-difference pressure correction equations apply to turbulentflow as well as to laminar flow. However, the momentum finite-difference equationsfor the turbulent flow have extra source terms than their laminar counterparts. TheNIMO higher-order scheme will now be applied to steady laminar flow in pipesand 2D steady laminar flow over a fence.

2.9.1 Steady laminar flow in pipes

Equations (50) and (51) are the governing equations for steady laminar flow in pipeswith constant physical properties. The exact analytical solution for this type of flowis a parabolic profile for the fully developed axial velocity distribution along theradial coordinate. It is given as:

u

u0= 2[1 − (r/R)2] (73)

where u0 is the uniform axial velocity at the pipe entrance, r is the radial distancefrom the pipe axis, and R is the pipe inner radius. Equation (73) is used here totest the accuracy of the NIMO scheme. Equations (52)–(59), (67), and (70)–(72)represent the NIMO finite-difference equations applicable to the laminar flow inpipes. The pipe radius R is taken as 1.0 cm and the inlet uniform axial velocity is1.0 m/s. The flowing gas is air at atmospheric pressure and 300 K. For this flow,Reynolds number is 1,300. The dimensionless pipe length (L/D) is 50. The pressurecorrection normal gradient, for each grid of the NIMO scheme, is equal to zero forthe four boundaries of the solution domain. This boundary condition is valid for allthe test problems reported below. Zero normal gradients of all dependent variables,at the exit section, are imposed. Again this boundary condition is common to alltest problems reported below. The inlet axial velocity is 1.0 m/s, while the radialvelocity is equal to zero. At the pipe wall the velocity components are equal to zero.However, at the pipe axis the axial velocity gradient is equal to zero, while the radialvelocity itself is equal to zero. For a uniform (80 × 80) grid, the number of iterationsis 1,000, which gives an error less than 0.1 percent in the finite-difference equations.Figure 2.22 depicts the radial profiles of the dimensionless axial velocity for axialdistances (x/D) = 2, 10, and 20. The axial velocity radial profiles for the main gridand the x-grid are consistent, showing continuous increase in the centerline velocity.The fully developed axial velocity radial profiles, shown in Figure 2.23 for the gridsof the NIMO scheme, are in excellent agreement with the exact analytical solutiongiven by equation (73). In Figures 2.22 and 2.23, um is the mean axial velocity. Theradial velocity profiles are depicted in Figures 2.24 and 2.25 for dimensionless axialdistances of 2, 10, 20, and 50. It is interesting to notice that the maximum negativeradial velocity occurs at (x/D) = 2; then it decreases sharply as the exit sectionis approached. Figure 2. 26 shows the variations in the pressure along the axialdistance. The numerical results indicate that the pressure is uniformly decliningalong the pipe length to overcome the laminar wall shear stresses.



0

1

2

0 1

1.5

0.5

0.2 0.4 0.6 0.8r /R

u/u

m

x /D = 2

x /D = 10

x /D = 20

Main grid

x-shifted

Figure 2.22. Dimensionless axial velocity profiles at different axial locations forthe laminar flow in pipes with Re = 1,300.

0

1

2

0 1

1.5

0.5

0.2 0.4 0.6 0.8r /R

u/u

m

x/D 50

Main grid

x-shifted

Analyticalsolution

Figure 2.23. Dimensionless axial velocity profile for laminar flow in a pipe withRe = 1,300.

2.9.2 Steady laminar flow over a fence

This is a 2D steady laminar flow over a single fence in a rectangular channel.The finite-difference equations used in the laminar pipe flow problem are alsoapplicable to flow over a fence. Excellent experimental results were obtained, usinglaser-Doppler anemometry, by Carvalho et al. [13]. The flow geometry used by



0

0 1

−0.002

−0.012

−0.01

−0.008

−0.006

−0.004

0.2 0.4 0.6 0.8r /R

v/u

m

x/D = 2

x/D = 10

Main gridx-shifted

Figure 2.24. Dimensionless radial velocity profiles at different axial locations forlaminar flow in pipes with Re = 1,300.

−0.00120 0.2 0.4 0.6 0.8 1

−0.001

−0.0008

−0.0006

−0.0004

−0.0002

0

r /R

v/u

m

Main grid

x-shifted

x/D = 20

x/D = 50

Figure 2.25. Dimensionless radial velocity at different axial locations for laminarflow in pipes with Re = 1,300.

Carvalho et al. [13] is shown in Figure 2.27. The inlet velocity is prescribed as afully developed profile for laminar flow between two parallel plates, namely,

ui/u0 = 1.5

[1 −

(H − 2y

H

)2]

(74)



−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

20 30 40 50 60

x/D

p −

p0

(N/m

2 )

Figure 2.26. Centerline axial pressure profile in the laminar flow in pipes withRe = 1,300.

H

y

x

5H 15H

0.1H

0.75

H

Figure 2.27. Geometry of flow over a fence.

where ui and u0 are the velocity along the x-axis at inlet and its mean value, respec-tively. The distance from the lower plate is denoted by y, while the vertical distancebetween the two plates is given as H . In the above equation, y ranges between 0.0 andH . The value of height H is equal to 1.0 cm, and u0 is equal to 0.175 m/s.The flowinggas is air at 300 K. The inlet Reynolds number (Re) is equal to 82.5, which is basedon u0 and the fence height S(= 0.75H ). The thickness of the fence is 0.1 cm. Theinlet flow velocity is computed from equation (74), while the transverse velocity (v)is equal to zero. Pressure correction normal gradients, u and v, are equal to zero onall solid surfaces of the flow domain. For this laminar flow problems two grids wereused, namely, (100 × 100) and (150 × 150). To obtain numerical errors less than 0.1



0

0.25

0.5

0.75

1

(a)x/S

y/H

Experimental data100 × 100 grid150 × 150 grid

−6.6 1.20−5.3

1.5

0

0.25

0.5

0.75

1

(b)

x/S

y/H

Experimetnal data100 × 100 grid150 × 150 grid

642

1.5

Figure 2.28. Comparison between predictions and measurements of dimensionlessvelocity along x-axis (u/u0) [13] for flow over fence, Re = 82.5.

percent, 1,757 and 4,000 iterations are used for the (100 × 100) and the (150 × 150)grids, respectively. This shows that the number of iterations for complete conversionis more than doubled in the case of the finer grid compared to the course one.

The transverse profiles of the dimensionless velocity component along the x-axisare depicted in Figure 2.28 for (x/S) = −6.6, −5.3, 0.0, 1.2, 2.0, 4.0, and 6.0. Theaxial distance is measured from the fence. For (x/S)< 0.0 the (100 × 100) and the(150 × 150) grids gave perfect fit to the experimental data. However, for (x/S)< 0.0the finer grid gave excellent agreement with the experimental data. However, gridsfiner than (150 × 150) could not give any higher accuracy. Figure 2.28 shows thatthe flow direction is downward for 0.0< (x/S)< 4.0; then the gas flows upwardlyfor higher values of the distance along the x-axis. Recirculation zones are thuscreated near the lower wall for (x/S)< 4.0 followed by recirculation zones at the



0

x /s−7

−1

−0.9

−0.8

−0.5

−0.4

−0.3

−0.2

−0.1

−0.6

−0.7

−3 109876543210−1−2−4−6 −5

0.1

P −

P0,

N/m

2

y/H = 0,150 × 150

y/H = 1,100 × 100

y/H = 1,150 × 150

y/H = 0,100 × 100

Figure 2.29. Axial pressure profiles near the upper and lower walls for the flowover a fence with 100 × 100 and 150 × 150 girds, Re = 82.5.

upper wall for higher values of (x/S). The radial pressure profiles along the x-axisare plotted in Figure 2.29 for the (100 × 100) and (150 × 150) grids. The profilesare chosen at (y/H ) equal to approximately 0.0 and 1.0, respectively, near the lowerand upper walls. It can be seen from this Figure that sudden pressure drop occursat the fence. However, it is smoother near the upper wall. Moreover, (100 × 100)grid predicts lower pressure at the fence with respect to the (150 × 150) grid. It isinteresting to notice the sudden increase in pressure as the flow changes its directionin the vicinity of (x/H ) ranging between 4 and 5.

2.10 Application of the NIMO Higher-Order Scheme to theTurbulent Flow in Pipes

For constant-density steady turbulent flow in pipes, the modeled time-meanmomentum equation can be written as follows:

∂(ρuiuj)

∂xi− ∂

∂xi

[(µt + µ)

∂uj

∂xi

]= − ∂p

∂xj+ ∂

∂xi

[(µt + µ)

∂ui

∂xj− 2

3ρkδij

](75)

where ui, uj, and p are time-mean values. The turbulent viscosity and turbulentkinetic energy are denoted by µt and k; in addition to time-mean momentumequations, the time-mean mass continuity must be included.

∂(ρui)

∂xi= 0 (76)



The turbulent viscosity can be computed with the help of the two-equationmodel of turbulence for the kinetic energy of turbulence (k) and its dissipation rate(ε). The governing equation for the kinetic energy of turbulence and its dissipa-tion rate are written here such that they can be applied to the standard k-ε model(Launder & Spalding [14]) and its low Reynolds number version of Launder andSharma [15].

∂

∂xi(ρuik) − ∂

∂xi

(µe

σk

∂k

∂xi

)= G − ρ(ε+ D) (77)

∂

∂xi(ρuiε) − ∂

∂xi

(µe

σε

∂ε

∂xi

)= (C1f1G − C2f2ρε)

ε

k+ ρE (78)

where G(=µt

∂uj

∂xi

[∂uj

∂xi+ ∂ui∂xj

])is the turbulent kinetic energy generation term and

is given by Launder and Spalding [14]. The turbulent viscosity µt is thus given as:

µt = Cµfµρk2/ε (79)

The high Reynolds number standard k-ε model is specified with the effec-tive viscosity µe =µt , D = E = 0.0, and f1 = f2 = fµ= 1.0. Moreover, the highReynolds number k-ε model logarithmic wall functions that prescribe at the wallthe shear stress, the generation term (G), and the value of ε are given by Launderand Spalding [14].

However, the low Reynolds number k-ε model of Launder and Sharma [15]takes the solution right to the pipe wall. The common constants between the twoversions of the k-ε model are given by Launder and Sharma [15] as cµ= 0.09,c1 = 1.44, c2 = 1.92, σk = 1.0, and σe = 1.3. Moreover, the low Reynolds numberadditional functions are defined as follows [15]:

fµ

= exp [−3.4/(1 + Rt/50)2] (80)

f1 = 1.0 (81)

f2 = 1 − 0.3 exp (−R2t ) (82)

D = 2(µ/ρ)

[∂(k)

12

∂r

]2

(83)

E = 2(µ/ρ)(µt/ρ)[∂2u

∂r2

]2

(84)

where the turbulent Reynolds number Rt is defined as [15]:

Rt = ρk2/(µε) (85)



Equations (83) and (84) are given specifically for the turbulent flow in pipeswhere r is the radial distance and u is the axial velocity.

In the low Reynolds number model, µe is defined asµ+µt and the wall bound-ary condition is defined as ∂p′/∂r = u = v= k = ε= 0.0. The boundary conditionat the centerline is defined as ∂φ/∂r = v= 0, where φ stands for all dependent vari-ables except v. At the exit section, the axial gradients of all dependent variables areequal to zero. At the inlet section, u = u0 and v= ∂p′/∂x = 0, where u0 is the inletuniform axial velocity.

The assessment of the accuracy of the results of the NIMO scheme is achievedwith the help of the experimental data of Laufer [16] at Re = 500,000 and the semi-empirical relation obtained by Schlichting [17] for a large number of experimentaldata for the fully developed turbulent pipe flow. The Schlichting 1/7th power law isgiven as [17]:

u/uCL =(

R − r

R

)1/70 < r < 0.96R (86)

where uCL is the fully developed centerline axial velocity.The numerical predictions for turbulent flow in pipes, using the NIMO higher-

order scheme, are shown in Figures 2.30–34. Air at 1.0 bar and 300 K enters thepipe at a uniform axial velocity of 29 m/s. The pipe diameter is 24.7 cm and itsdimensionless length (L/D) is equal to 30. For this flow, the Reynolds number is500,000. For the low and high Reynolds number turbulent models, a (100 × 100)grid is used. The grid of the high Reynolds number model is uniform, while thatfor the low Reynolds number model converses toward the pipe wall. For both modelsthe number of iterations approximates 1,000, giving errors less than 0.1 percent inthe difference equations.

The developing turbulent flow in pipes is depicted in Figure 2.30 for axialdistances (x/D) equal to 2 and 5. The dimensionless axial velocity (u/um) numericalresults of the main grid are essentially indistinguishable from those of the x-grid,as can be seen from Figure 2.30. The near-wall region is shown in Figure 2.30b,to facilitate the comparison of the wall effect as predicted by the low and highReynolds number models. The wall effect of the low Reynolds number modelpropagates into the core of the turbulent flow in pipes, while the high Reynoldsnumber model has its wall effect restricted to (r/R)> 0.95 for (x/D)< 5. Moredeveloping flow numerical results are shown in Figure 2.31 for (x/D) equal to10 and 20. The high Reynolds number k-ε model wall effect is still confined to(r/R)> 0.75, while the low Reynolds number counterpart has its wall effect allover the turbulent flow for (x/D) = 20. The dimensionless radial velocity (v/um)profiles are depicted in Figure 2.32, at (x/D) = 5 and 20 and at the exit section.The low Reynolds number model predicts higher absolute radial velocities, whileboth models predict a vanishing radial velocity at the fully developed flow near theexit section of (x/D) = 30. The fully developed turbulent time-mean axial velocityradial profiles are depicted in Figure 2.33, for the high and low Reynolds numbermodels. On the same figure, the measured fully-developed dimensionless axial



0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Main grid, x /D = 2, lowRex-shifted, x /D = 2, low ReMain grid, x /D = 5, lowRex-shifted, x /D = 5, low ReMain grid, x /D = 2, high Rex-shifted, x /D = 2, high ReMain grid, x /D = 5, high Rex-shifted, x /D = 5, high Re

r /R

u/u

m

0

0.2

0.4

0.6

0.8

1

1.2

0.75 0.8 0.85 0.9 0.95 1

r /R

u/u

m Main grid, x /D = 2, low Rex-shifted, x /D = 2, low ReMain grid, x /D = 5, low Rex-shifted, x /D = 5, low ReMain grid, x /D = 2, high Rex-shifted, x /D = 2, high ReMain grid, x /D = 5, high Rex-shifted, x /D = 5, high Re

(a)

(b)

Figure 2.30. Dimensionless axial velocity (u/um) at different axial locationsfor turbulent flow in pipes with Re = 500,000. (a) r/R from 0 to 1.(b) r/R from 0.75 to 1.

velocity [16] and the similar profile that can be calculated by the semi-empiricalrelation of Schlichting [17] are also plotted. The NIMO scheme predicts excellentagreement with the fully developed axial velocity profile given by the 1/7th powerlaw of Schlichting [17] when the low Reynolds number k-εmodel is used. However,the present numerical results for u/uCL fall nicely between the experimental dataof Laufer [16] and the Schlichting’s 1/7th power law [17]. On the other hand, thehigh Reynolds number model poorly predicts the fully developed turbulent pipeflow and hence should not be adopted for this type of flow. Finally, the pressureaxial distributions, predicted using the low and high Reynolds number k-εmodels,are shown in Figure 2.34. This figure shows clearly that the absolute axial pressure



0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1r /R

u/u

m Main grid, x/D = 10, low Re

x-shifted, x/D = 10, low Re

Main grid, x/D = 20, low Re

x-shifted, x/D = 20, low Re

Main grid, x/D = 10, high Re

x-shifted, x/D = 10, high ReMain grid, x/D = 20, high Re

x-shifted, x/D = 20, high Re

Figure 2.31. Dimensionless axial velocity at different axial locations for turbulentflow in pipes, Re = 500,000.

−0.004

−0.003

−0.002

−0.001

0.000

0 0.2 0.4 0.6 0.8 1r/R

v/u

m Main grid, x/D = 5, low Rex-shifted, x/D = 5, low ReMain grid, x/D = 20, low Rex-shifted, x/D = 20, low ReMain grid, exit, low Rex-shifted, exit, low ReMain grid, x/D = 5, high Rex-shifted, x/D = 5, high ReMain grid, x/D = 20, high Rex-shifted, x/D = 20, high ReMain grid, exit, high Rex-shifted, exit, high Re

Figure 2.32. Dimensionless radial velocity profiles at different axial locations forturbulent pipe flows, Re = 500,000.



0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1r/R

u/u

CL

Numerical data, low ReNumerical data, high RePower lawExperimental data

Figure 2.33. Dimensionless time-mean axial velocity for turbulent pipe flows usinglow and high Reynolds number models, Re = 500,000.

−300

−250

−200

−150

−100

−50

0

0 10 20 30

x/D

P −

P0

(N/m

2 )

Low ReHigh Re

Figure 2.34. Centerline axial pressure profiles for turbulent flow in pipes,Re = 500,000.

gradient of the low Reynolds number model is approximately five times higher thanthat of the high Reynolds number model. Alternatively, the wall shear stress in thelow Reynolds number model is five times that of the high Reynolds number k-εmodel.



2.11 Conclusions

This chapter describes a novel numerical procedure–the NIMO finite-differenceprocedure. The NIMO numerical procedure is free from any interpolations of con-vective fluxes that normally produce high levels of false diffusion in most knownfinite-difference schemes. The NIMO system involves four grids in 2D flows. Thefour grids are located in space such that each grid is displaced midway with respectto one of the remaining grids. The main grid and the xy-grid interconnect indi-rectly through the x- and y-grids and vice versa. Similar interconnection betweenthe four NIMO grids is valid in the cylindrical-polar solution domain. The percent-age sum of the absolute residual errors of the main grid and xy-grid behaves inthe same fashion, which is different from the similar trend of the other two grids.The NIMO scheme is applied to two test problems. The first test problem is theoblique flow in the Cartesian coordinates, while the second test problem is repre-sented by a rotating radial flow in an annular sector. The NIMO finite-differencescheme can produce numerical profiles of φ that capture the analytical solution ofthe test problem for 45-degree and 30-degree oblique flows for high Peclet num-bers. Moreover, NIMO can essentially capture the analytical solution of the twotest problems for square, rectangular, and nonuniform grids. The second test prob-lem is also computed with a very high accuracy, which confirms the validity ofthe NIMO scheme to the cylindrical-polar coordinates as well as to the Cartesiancoordinates. Most of the single-grid numerical procedures utilizing rectangular ornonuniform grids could not reach that level of accuracy due to false diffusion,overshooting, or undershooting. This could be concluded from the comparison ofthe data of these schemes with the corresponding present numerical results. TheNIMO finite-difference equations contain convective terms that slightly resemblethe upwind schemes. However, unlike the upwind scheme, the incoming convectivefluxes to a particular grid are supplied by two of the remaining three grids of NIMO.A new feature emerges that allows the outgoing fluxes to leave behind positive ornegative traces in a particular grid control volume, which is completely ignoredin the upwind scheme. These outgoing convective fluxes constitute the incomingconvective fluxes to two of the remaining three grids of NIMO. Results obtained fordifferent values of the Peclet number showed that the diagonal core of the test prob-lem is surrounded by narrow strips bounded by φ= 0.99 and 10−2. These narrowstrips become essentially nonexistent as the Peclet number is increased to 1,000.Theconverged numerical solutions of the four NIMO grids for a passive scalar variableat a fixed value of the Peclet number are consistent and identical. The NIMO higher-order scheme has further been tested against laminar and turbulent flow in pipesas well as recirculating flow behind a fence. The laminar flow in pipes is predictedwith a very high accuracy for a (100 × 100) grid. However, the recirculating laminarflow behind a fence can be predicted accurately with a finer (150 × 150) grid. Theturbulent flow in pipes is well predicted when the low Reynolds number k-e modelis used.



Nomenclatures

A surface area of control volumea finite-difference coefficientsc convective transport ratesc, c modified convective transport ratescµ, c1, c2, σk , σe common coefficients in high and low Reynolds

number modelsD, E, f1, f2, fµ coefficients in low Reynolds number modeld diffusive transport ratek turbulent kinetic energyn number of iterationsPe Peclet numberp′ pressure correctionRt turbulent Reynolds numberSu, Sp source term coefficients in linear formu velocity component along x directionuCL fully developed centerline axial velocityuk fluid velocity along coordinate direction xiv velocity component along y directionxk coordinate directionδx distance between central node and neighboring nodeε dissipation rate of turbulent kinetic energyφ scalar variableφ molecular diffusivity of φµe effective viscosityµt turbulent viscosityρ density

Superscript

E, W east and west directionsp value at center of control volumeS, N south and north directionst turbulent flowi, j grid node indices

Superscript

x x-gridy y-gridxy xy-grid* uncorrected values



References

[1] Abou-Ellail, M. M. M., Gosman, A. D., Lockwood, F. C., and Megahed, I. E. A.Description and validation of a three-dimensional procedure for combustion chamberflows. AIAA Journal of Energy, 2, pp. 71–80, 1978.

[2] Patankar, S. V. Numerical Heat Transfer and Fluid Flow. McGraw Hill: New York,1980.

[3] Roache, P. J. Computational Fluid Dynamics. Hermosa: Albuquerque, NM, 1972.[4] Prince, H. S., Varga, R. S., and Warren, J. E. Application of oscillation matrices to

diffusion-convection equations. Journal of Mathematics and Physics, 45, pp. 301–311,1966.

[5] Leonard, B. P. A survey of finite differences of opinions on numerical modeling ofincomprehensible defective confusion equation. ASME Applied Mechanics Division,34, pp. 59–98, 1979.

[6] Leonard, B. P. Stable and accurate convective modeling procedure based on quadraticupstream interpolation. Computer Methods in Applied Mechanics and Engineering, 19,pp. 59–98, 1979.

[7] Leonard, B. P. A stable, accurate, economical and comprehendible algorithm for theNavier–Stokes and scalar transport equations. In: Numerical Methods in Laminarand Turbulent Flow, C. Taylor and B. A. Schrefler, eds., Proceedings of the SecondInternational Conference, pp. 543–553. Venice, Italy, 1981.

[8] Song, B., Liu, G. R., Lam, K. Y., and Amano, R. S. Four-point interpolation schemesfor convective fluxes approximation. Numerical Heat Transfer, Part B, 35, pp. 23–39,1999.

[9] Song, B., Liu, G. R., Lam, K. Y., and Amano, R. S. On a higher-order boundeddiscretization scheme. International Journal for Numerical Methods in Fluids, 32, pp.881–897, 2000.

[10] Song, B., Liu, G. R., and Amano R. S. Applications of a higher-order bound numericalscheme to turbulent flows. International Journal for Numerical Methods in Fluids, 35,pp. 371–394, 2001.

[11] Raithby, G. D. Skew upstream differencing schemes for problems involving fluid flow.Computer Methods in Applied Mechanics and Engineering, 9, pp. 153–164, 1976.

[12] Verma, A. K., and Eswaran, V. Overlapping control volume approach for convection-diffusion problems. International Journal for Numerical Methods in Fluids, 23,pp. 865–882, 1996.

[13] Carvalho, G. M., Durst, F., and Pereira, C. F. Predictions and measurements of laminarflow over two-dimensional obstacles. Applied Mathematical Modeling, 11, pp. 23–34,1987.

[14] Launder, B., and Spalding, D. The numerical computation of turbulent flows. ComputerMethods in Applied Mechanics and Engineering, 3, pp. 269–289, 1974.

[15] Launder, B., and Sharma, B. Application of the energy-dissipation model of turbulenceto the calculation of flow near a spinning disc., Letters in Heat and Mass Transfer, 1,pp. 131–137, 1974.

[16] Laufer, J. The structure of turbulence in fully developed pipe flow. NACA Report no.1174, 1954.

[17] Schlichting, H. Boundary-Layer Theory, Chapter 20, McGraw-Hill: New York, 1979.

Sunden ch003.tex 27/8/2010 18: 35 Page 61

3 CFD for industrial turbomachinerydesigns

C. Xu and R. S. AmanoUniversity of Wisconsin, Milwaukee, WI, USA

Abstract

An implicit method was introduced to solve the viscous turbulent flow in a time-efficient way. This chapter presents how to use the numerical method for applyingthe airfoil blade design improvements in turbomachinery. This step includes bladeaerodynamic design system with optimization approaches. The design methoddeveloped by the authors are introduced that employs a discrete vortex methodto calculate the blade pressure distributions after the airfoil section was designed,and a fast N-S solver developed in this study was used to further check the bladeperformance. With the approaches described in this chapter, the whole blade aero-dynamic design procedure can provide a fast way to design the blade profile thatmeets the blade design requirements. The present Navier-Stokes (N-S) code canwell handle both subsonic and supersonic flows, which can be further connected toan optimization code to performance three-dimensional blade design optimizations.

Keywords: CFD, Compressor, Finite-volume method, Implicit method,Optimization, Turbomachinery

3.1 Introduction

3.1.1 Computational methods in turbomachinery

Flows in turbomachinery are among the most complex flows encountered in fluiddynamic practices as in most cases, they are three dimensional, with all three flowregimes (laminar, transition, and turbulent) with flow separations. Flows may beincompressible, compressible, subsonic, transonic, or supersonic; some turboma-chinery flows include all of these flow regimes. Equations are strongly coupled andcomplex boundary conditions are often unavoidable. A large number of flow andthermal parameters are encountered: Reynolds number, Mach number, rotationalspeed, Prandtl number, and flow incidence, with highly complex geometrical param-eters. This complex flow and geometrical variables dictate the nature of governingequations and flow solvers to be used.



Full N-S

Physical approximation

Procedure

CoupledUncoupled

• Relaxation pressure based• Pseudocompressible method• Time marching

• Elliptic for cross flow• Parabolic for steamwise

• Grid–free method• Euler method• N-S method

Elliptic N-SParabolic N-S

Pressureapproximation

Velocitysplit

RNS, TLNS,PPNS

Figure 3.1. Classifications of governing equations.

With the rapid development of computer hardware, the use of computa-tional fluid dynamics (CFD) in turbomachinery design and analysis has beenever increasing. Aerodynamic processes associated with the flow of fluid throughturbomachines pose one of the toughest challenges to the computational fluiddynamicists. Although with the recent development of computer performance sci-entists and engineers can look more and more flow phenomena in turbomachinedesigns, applications of full three-dimensional multistage simulations in the aero-dynamic design are still a challenge and so is, the design of effective schemes thatcan be used for turbine design [1–6].

There are several types of approximations in the numerical solution of gov-erning equations for turbomachinery flow simulations. Governing equations areapproximated depending on the physics of flowfield. Further approximationsare in numerical discretization and technique in solving these equations. Theclassifications for governing equations are shown in Figure 3.1. The numericalprocedures classified according to the physical approximations made are inviscidflows, parabolized Navier–Stokes techniques, and thin-layer Navier–Stokes andfull Navier–Stokes solutions. According to the flow equations to be solved, thenumerical method can be categorized into three types: i.e. grid-free vortex method[7–14], incompressible Euler equation and Navier–Stokes simulation method [2–6],and compressible Euler equation and Navier–Stokes simulation method [14–36].Discussions of the numerical methods here are mainly on the Navier–Stokesequations. Brief discussions of the grid-free vortex method and incompressiblesimulation method are also provided. The main focus will be on the discussions ofthe compressible simulation method with emphasis on the Navier–Stokes method,which is directly related to the present study.


CFD for industrial turbomachinery designs 63

3.1.2 Grid-free vortex method

Vortex methods are based on the frequency-domain source-doublets and vortex pan-els that can predict steady and unsteady flows in turbomachines. In this approach,airfoils of a turbomachine are discretized in two several vortex panels and source-doublets to represent boundaries. Vortex methods can handle not only steady flowsbut also unsteady flows [7–14]. Most of these methods are based on the inviscidassumption. The vortex methods are still limited in irrotational flow, and they stillhave some difficulties in handling heat transfer problem. A grid-free vortex methodcan be adapted and extended to use as part of blade design procedure. In the vortexmethod, the rotor/stator stage is decoupled into two separate blade rows. Each bladerow interacts only with the immediate downstream or upstream of a blade row. Theinfluences from other blade rows are assumed to be much smaller than that fromthe blade from the immediate neighboring blade rows. The wake profile can be ananalytical or measured time or spatially dependent velocity distribution. The veloc-ity distributions are generally not the simple harmonic type. Harmonic amplitudesand related phase angles of this nonsimple harmonic wake-field can be obtainedthrough a Fourier decomposition of the velocities. The wakes of the blades are usu-ally calculated based on some empirical wake models. The unsteady response onthe downstream blade row is calculated in a typical blade-wake interaction manner.This method can predict the interactions between blades and blades as well as bladesand wake. The panel method can predict the pressure distribution accurately. Thecomputational time using this method is more than 100 times faster than a typicalNavier–Stokes solution. This method can be coupled to a boundary layer code topredict viscous effects and losses. The main limitations of this method are based onthe assumption that flow is irrotational and incompressible. The three-dimensionalturbine flows are almost invariably rotational, and the existence of potential flow isan exception.

The detailed formulation is described in Refs. [7–10]. The flow around a tur-bine airfoil can be represented by vortex sheets and/or source sheets of unknownstrength clothing the whole airfoil. The location of the sheet is measured clockwisearound the body perimeter from the trailing edge. The airfoil profile is dividedinto N planar elements denoted by (xq, yq) (where q = 1, 2, . . . , N ) with elements 1and N being adjacent to the trailing edge. The calculation assumes that a uniformvortex distribution of an initially unknown vorticity is placed on each element q.The Biot–Savart law provides a means of calculating the velocities in the flow field.The boundary condition on the airfoil surface is such that the resulting flow mustbe parallel to the body surface at all control points.

The condition of a zero internal tangential velocity at the pth collocation point(where p = 1, 2, . . . , N ) is given by

(Q + Q∞) • τ|p = 0 (1)

where Q is the velocity induced by each vortex element. While the free-streamtangential velocity component can be calculated directly, it can be moved to the



right-hand side of equation (1) and be given as follows:

(Q • τ)|p = −(Q∞ • τ)|p = −U∞ cosβp − V∞ sin βp (2)

where U∞ and V∞ represent the two components of uniform stream velocityparallel to the x- and y-axes, respectively. βp is the slope of the body profile and isdefined by

βp = tan−1(

yp − yp−1

xp − xp−1

)(3)

Letlq be the length of the qth element. On using the boundary condition, equation(1) can be expressed as a set of linear equations

N∑q=1

Kpq γq = −U∞ cosβp − V∞ sin βp (4)

where Kpq denotes the coupling coefficients linking p and q and is given by

Kpq = lq2π

[(yp − yq) cosβp − (xp − xq) sin βp]

[(yp − yq)2 + (xp − xq)2](5)

After the vortex strength of each element is calculated, the velocity vector fieldand velocity potential field can be obtained. The calculated velocity potential andvelocity can then be used to obtain the pressure and the isentropic Mach numberdistribution. To validate the code, a NACA 0012 airfoil with 4-degree incidencewas used.

3.2 Numerical Methods for Incompressible Flow

From the numerical method point of view, it is needed to point out that the techniquestypically used for the solution of Euler and Navier–Stokes equations are the same,even though there are some differences in the behavior of the technique with andwithout the viscous terms. The techniques typically used by both Euler and Navier–Stokes equations are discussed in the following sections. To predict and improvethe performance of turbomachine, it is important to understand the flow and heattransfer through the turbine blade and blade passages from the standpoint of thedesign of blade geometry. For the flow inside the turbine stages from the firststage to the last one, the flow character may change from incompressible potentialflow to compressible flow. The incompressible solution is also an important partof the turbomachinery flow simulations. In most of the early studies [6,10], time-average incompressible flow equations were considered. In those methods the firstand second derivative terms did not contain cross-derivatives that can be treated



implicitly. The high-order artificial dissipation terms can be used in the calculationin order to increase the accuracy when the solutions meet the stability requirement.To calculate the pressure, those schemes need to incorporate the pressure correctionequations.

Pressure-based methods are one of the major methods for incompressible flowanalysis. In this method, the assumed pressure is updated using an auxiliary equa-tion for pressure, and a multipass procedure is adapted to help converge both thepressure and the velocity field. Most of the pressure-based methods are exten-sions of the single-pass pressure correction methods. In this technique, a separateequation for pressure correction is developed for both the bulk pressure correc-tion and the pressure correction in the transverse momentum equation, relatingthe source term in the Poisson equation to the pressure correction in the veloc-ity field through the continuity equation (10). The pressure correction requiresa staggered grid to eliminate oscillations in the solution of the pressure correc-tion equation. Both finite-difference and finite-volume technique can be usedin these methods. Application of these techniques to turbomachinery flows andsubsequent modifications to improve accuracy and efficiency have been carriedout. The author has modified and improved convergence of the pressure correc-tion method [10]. The pressure-based methods are one of the most sophisticatedmethods. They are highly recommended for incompressible turbomachinery flowsimulation.

3.3 Numerical Methods for Compressible Flow

For the numerical method for compressible flow, the time-marching technique isone of the most important methods. The principle of a time-marching method is toconsider the solution of a stationary problem as the solution after a sufficiently longcalculation time of the time-dependent equations describing the particular problem.The attraction of the time-marching method is the ability to compute mixed subsonicand supersonic flows. The equations may be solved by either finite-difference orfinite-volume form. In the former, it is usual to transform the computational domaininto a uniform rectangular grid and to express the derivatives of the flow variables interms of values at the nodes of this grid. In the finite-volume method the equationsare regarded as these for the conservation of mass, energy, and momentum applied toa set of interlocking control volumes formed by a grid in the physical plan. Whenequations are solved in this way, it is easier to ensure conservation of mass andmomentum than in the differential approach, but numerical schemes are necessaryto ensure stability.

The advantages of the time-marching methods are as follows: (a) the same codecan be used for the solution of all flow regimes: subsonic to supersonic, viscous orinviscid equations; (b) the technique is easy to implement and vectorize to take anadvantage of supercomputing capabilities. The equations are solved as a coupledsystem. The scalar variables (pressure, velocities, density, enthalpy, entropy, andtemperature) are solved simultaneously; (c) the code is flexible so that it can employ



both external and internal flow calculations; (d) for complex flow geometry andflow fields, the technique is comparable in other methods. The major disadvantagesare as follows: (a) since the technique is time iterative and often requires very smalltime steps, the computational efficiency is low. This shortage stimulates the authorto develop a new kind of economical time-marching algorithm; (b) the techniqueis not suitable for incompressible flows. This also stimulates the author to solve theeigenvalue stiffness problem.

Since the flows in turbomachinery components contains both incompressibleand compressible parts, the development of the methods that can be used for bothincompressible and compressible flows is very important. The pressure-based meth-ods solve a Poisson equation for pressure, and the momentum equations in anuncoupled manner by iterating until a divergence-free flow field is satisfied. Onthe other hand, it is evident from the material presented in the earlier section thatvery efficient and accurate algorithms have been developed for solving hyperbolicsystems governing compressible flow using time-iterative methods. Many attemptshave been made to combine these two methods and hope to form a kind of newmethod that can be used in both impressible and compressible flows.

Two methods have been proposed. One involves recasting the incompressibleflow equation as hyperbolic systems, which is called pseudo-compressibility tech-nique. The first method is to premultiply the time derivative by a suitable matrixand the other is to use a perturbed form of the equations in which specific termsare dropped such that the physical acoustic waves are replaced by pseudo-acousticmodes. This technique is called preconditioning method [6].

Pseudo-compressibility technique is based on making the governing equationsbehave like a hyperbolic system through adding the time derivative of a pressureterm. In the steady state, these equations and solutions are exactly those of steadyincompressible flow, even though the temporal solutions are not accurate. The conti-nuity equation after modification introduces spurious pressure waves of finite speedinto the flowfield through the pseudo-compressibility parameter. Most researcherslinearized the equations using a truncated Taylor series and discretized them byusing a two-point central difference in the spatial directions. The most popularsolution techniques are the standard implicit approximate factorization schemes.The pseudo-compressibility technique has been used in the turbine blade flow cal-culation. However, the pseudo-compressibility parameter is difficult to determine.It is strongly influenced by the flow structure.

The preconditioning method, on the other hand, is based on modifying the time-dependent Navier–Stokes equations. The preconditioning matrix was added in thetime derivative term. The basic idea of the preconditioning procedure is to startwith the time derivatives expressed in terms of the viscous dependent variables thatappear in the diffusion terms.The procedures modify the eigen structure of the invis-cid components of the set of the Navier–Stokes equations such that the conditionnumber remains bounded as the freestream Mach number becomes small. Providedthat the numerical discretization is modified to reflect the new eigen system, the useof time-derivative preconditioning can allow a single algorithm to provide accurate,efficient solutions to the Navier–Stokes equations at all flow speeds.



For obtaining the stable scheme, most of the researchers used the implicit schemeto yield a promising method for rapid calculations of fully coupled flows at allspeeds. However, the selection of the preconditioning matrix is according to expe-rience and the flow characteristics. The limitations of current methods encourageus to investigate the new methods. Authors [2 and 3] proposed a timing matchingmethod through modifying the governing equations to prevent eigenvalue stiffnesswhen it uses in incompressible flow range.

There are two classes of methods for solving the time-dependent equations:explicit and implicit. Explicit methods, where spatial derivatives are evaluatedusing the known conditions at the old time level, are simpler and more easilyvectorizable. The implementation of the boundary conditions is straightforward.The coding can be easily extended to include a time-dependent calculation. Theirmajor disadvantages lie in the conditional stability dictated by courant-Friedrichs-lewy limit. The convergence is usually slower and requires more computationaltime than an implicit method. For implicit methods, on the other hand, the unknownvariables are derived from a simultaneous solution of a set of equations. Implicitmethods usually allow larger time steps and faster convergence, and are attractivefor both steady and unsteady flows. Both explicit and implicit schemes are widelyemployed in the computations of turbomachinery flows. However, both of thesemethods are still under development.

The most widely used explicit methods in turbomachinery flow are Lax–Wendroff scheme, MacCormack scheme, and Runge–Kutta scheme [6]. TheRunge–Kutta scheme allows larger time step than that with other schemes. Artificialdissipation must be purposefully added to these schemes, usually after each stage inthe updating procedure. All of these schemes provide inherent artificial dissipationdue to truncation error. The most widely used implicit techniques in turbomachin-ery flow are approximate factorization and upwind schemes. The literature showsthat operational CFL number in the order of 10 for viscous turbomachinery flowcalculations provided the best error damping properties although linear stabilityanalysis provided that the schemes are unconditionally stable in two dimensions[1–3]. The implicit scheme enables large time steps but may introduce factorizationerror. The results for a turbine cascade indicate that all techniques provide nearlyidentical pressure distribution. The major differences lie in the CFL number andCPU time for convergence. Both explicit and implicit techniques are widely used,with explicit techniques more commonly used in industry. Both techniques canincorporate local time stepping. The maximum time step for a four-stage Runge–Kutta method is CFL = 2.8, and typical CFL numbers for the implicit lower upperscheme and alternating direction implicit scheme are between 5 to 10, [6]. There-fore, the choice of explicit versus implicit technique and the time accuracy of agiven algorithm may have less to do with the accuracy of the predicted results thanthe nature of a numerical damping and an application grid.

In the turbomachinery flow field simulation, most researchers [20–32] focusedon the flow field analysis. In the flow and heat transfer analysis, the governingequations may be solved in either finite-difference or finite-volume form. Thefinite-difference scheme usually provides a good convergent rate. The finite-volume



scheme provides a very simple and automatic conservation of mass and momentumscheme, and it also has good physical properties because the scheme is obtainedfrom working in a physical grid. The argument as to whether finite-difference orfinite-volume schemes are preferable is not resolved, and both types are still widelyused. Either scheme is written in the form of finite-difference scheme. The finite-difference equations use to simulate flow and heat transfer must be numericallystable, spatially and temporally accurate and efficient, conservative so that flowdiscontinuities can be removed from every grid without any distortion, and easilyapplicable in generalized coordinates.

Aerodynamic designers have been using the CFD approach as one of the designtools to generate engineering design data. The potential of CFD approaches torevolutionize a turbomachinery analysis is exemplified by impressive calculationssuch as those of flows around turbine stators and rotors. A general method forCFD in the design of turbomachinery, which would include turbulence viscouseffects with minimum approximations and high efficiencies, could provide a majortool for both researchers and design engineers to improve turbine performance anddurability. One of the motivations of this part has led to a major effort in developingcascade flow analysis of various degrees of sophistication. Despite significant gainsin the computational speed of modern computers, most numerical schemes stillrequire prohibitive amounts of computational time to obtain solutions of the Navier–Stokes equations. Despite the challenges of the three-dimensional complicatedflow simulation, a two-dimensional flow simulation is still an interesting topic of aresearch scientist [1,4].

Although considerable progress has been made in the past 20 years in the numer-ical simulation of steady two-dimensional turbine cascade flows, these numericalresults were still in general not reliable enough [1]. Further improvement is stillneeded in the turbulence modeling, grid generation, and efficiency of the numericalscheme. The fact is that turbulence models themselves have not always performedwell. The choice of the numerical method might also be a crucial factor for thesuccessful implementation of a numerical scheme. The present study demonstratesthe efficiency of the numerical scheme and potential usage of this scheme to solvecascade flow and heat transfer problems.

Time-marching algorithms [2,14] have been used primarily in gas turbine cas-cade flow and heat transfer analysis. The basic principle of a time-marching methodis to consider the solution of a stationary problem as the solution after sufficientlylong calculation time of the time-dependant equations. The computation starts witha rough perturbation that develops under certain boundary conditions until it reachesa convergent state. In this approach, the governing equations are replaced by arobust time-difference approximation with which steady or time-dependent flowsof interest can be solved. At each time level, a system of algebraic equations issolved. The time-marching algorithms can be categorized into two types: explicitand implicit schemes. An explicit method, in general, is easier to program and tovectorize, allowing incorporation of boundary conditions and turbulence models ina simpler way than an implicit method. A main shortcoming of the explicit methodis its susceptibility to numerical stability, which constrains the time-step size rather



severely. The implicit scheme, on the other hand, is generally unconditionally stable,so that a relatively large time step can be adapted. In addition to these algorithms,there exists a family of hybrid algorithms, which is widely used in the cascade-flowsimulations. The hybrid algorithms possess the positive features of the explicit andimplicit algorithms, providing a relatively rapid convergence process and havinga less restricted stability constraint. The essence of this algorithm is the use ofexplicit and implicit finite-difference formulae at alternative computational meshpoints. However, the nature of the basic formulae for the hybrid algorithm remainsthe same as for the traditional implicit and explicit schemes.

In viscous calculations, dissipating properties always present due to the exis-tence of diffusive terms. Away from the shear layer regions, however, the physicaldiffusion is generally not sufficient to prevent the numerical oscillation of theschemes. Thus, the artificial dissipation terms are usually used in order to sta-bilize and permit a higher CFL number in most of the interactive schemes for thecompressible flow computation. As is commonly recognized, the artificial dissipa-tion will influence the accuracy of the numerical results. It has been shown thatessentially grid-converged solutions for the high-speed viscous flows over aerody-namic shapes can be obtained if sufficiently fine meshes for the computation areprovided. However, the computation based on fine meshes is very time consuming,making it difficult to assess the numerical accuracy of grid-converged solutions.The techniques to properly treat dissipation terms are still an important topic of thecomputational fluid. An attempt was made by Turkel and Vatsa [15] to improve theaccuracy of the solutions on a given grid in order to reduce the required numberof grid points for obtaining a specified level of accuracy. The essential mechanismused in their method [15] is to reduce the level of the artificial viscosity by replacingthe scalar form of the artificial viscosity with a matrix form. It has been shown thatthe numerical accuracy of the Navier–Stokes solutions is improved through the useof the matrix-valued dissipation model. In this chapter, the dissipation terms werenot treated through modifying convective fluxes as in traditional methods. Instead,the dissipation terms were incorporated into the time-derivative terms to form a newtime discretization scheme to reduce the level of artificial viscosity. And also in thisway, the eigenvalue-stiffness problem associated with the time-derivative term canbe prevented when the calculations are performed in a low Mach number flow. Thischaracteristic can partly control the effect of eigenvalue stiffness, which is wellknown on the convergence of both explicit and implicit schemes. Furthermore, thepresent scheme is a hybrid scheme that combines the advantages of the implicitand explicit schemes, and is a second-order time and spatial derivative scheme thatallows more accurate calculation. Because the higher CFL number can be used withthe present method, it is more economic than the ordinary implicit scheme.

The effect of eigenvalue stiffness on the convergence of both explicit and implicitschemes is well known, and normally two distinct methods have been suggestedfor controlling the eigenvalues to enhance convergence. There are two kinds ofmethods that can solve this problem. One of the methods to overcome this problemis to premultiply the time derivative by a suitable matrix, and the other is to usea perturbed form of the equations in which specific terms are dropped such that



the physical acoustic waves are replaced by pseudo-acoustic modes. The presentmethod follows the first idea and incorporates the matrix-valued dissipation termsin time-derivative terms. These terms prevent the eigenvalues of the time-derivativeterms from becoming stiff when the calculations are performed in the flow wherethe Mach number is relatively small. This characteristic can partly control the effectof eigenvalue stiffness that is well known on the convergence of both explicit andimplicit schemes. Furthermore, the present scheme employs a hybrid algorithmthat combines the advantages of the implicit and explicit schemes. The presentscheme also processes a spectral radio technique to simplify the calculation andavoid the approximate factorization, thus increasing the stability. This new methodis a second-order time and spatial derivative scheme permitting a high CFL number.

3.4 Governing Equations for Two-Dimensional Flow

The two-dimensional time-dependent Reynolds-averaged Navier–Stokes equationscan be written as:

Qt + Ex + Fy = Rx + Sy (6)

where

Q =

⎡⎢⎢⎣ρ

ρuρv

e

⎤⎥⎥⎦ , E =

⎡⎢⎢⎣ρu

p + ρu2

ρuv(e + p)u

⎤⎥⎥⎦ , F =

⎡⎢⎢⎣ρv

ρuvp + ρv2

(e + p)v

⎤⎥⎥⎦ , R =

⎡⎢⎢⎣0τxxτxy

βx

⎤⎥⎥⎦ , S =

⎡⎢⎢⎣0τxyτyy

βy

⎤⎥⎥⎦(7)

and where

τxx = 2µux + λ(ux + vy)

τxy = µ(ux + vy)

τyy = 2µuy + λ(ux + vy)

βx = uτxx + vτxy + γµP−1r ex

βy = uτxy + vτyy + γµP−1r ey

e = p/[ρ(γ − 1)]

λ = −2µ/3

µ = µt + µl

The equation of state for a perfect gas is:

p = ρRT (8)



And the definition of the static enthalpy supplementing the equations of motion canbe written as:

h = CpT (9)

Because the turbine cascade flow always accompanies high temperature, the flowproperties change with the temperature variation. In this calculation, Sutherland’sLaw is used to express the relation of molecular viscosity and temperature:

µ = µ0

[(T

T0

)1.5 T0 + S

T + S

](10)

where S = 100 K for air and the variation of thermal conductivity with temperatureis determined by:

k = k0

[(T

T0

)1.5 T0 + S ′

T + S ′

](11)

where S ′ = 194 K for air. Making the variable transformation for the governingequations from physical domain (x, y) to computational domain (ξ, η), we have

ξ = ξ(x, y) (12)

η = η(x, y) (13)

The Navier–Stokes equation can be written as:

Qt + Eξ + Fη = Rξ + Sη (14)

where

Q = 1

J

⎡⎢⎢⎣ρ

ρuρv

e

⎤⎥⎥⎦ , E = 1

J

⎡⎢⎢⎣ρu

pξx + ρuupξy + ρuv(e + p)u

⎤⎥⎥⎦ , F = 1

J

⎡⎢⎢⎣ρv

pηx + ρuvpηy + ρvv(e + p)v

⎤⎥⎥⎦

R = 1

J

⎡⎢⎢⎣0

τxxξx + τxyξyτxyξx + τxyξyβxξx + βyξy

⎤⎥⎥⎦ , S = 1

J

⎡⎢⎢⎣0

τxxηx + τxyηy

τxyηx + τyyηy

βxηx + βyηy

⎤⎥⎥⎦where the contravariant velocities are given by:

u = uξx + vξyv = uηx + vηy



Equation (14) can be written in the similar form to equation (6) as:

Qt + Eξ + Fη = Rξ + Sη (15)

A satisfactory turbulence model should meet the requirements of accuracy andcompatibility [1–5]. Although the flow around the turbine cascade is extremelycomplicated, the boundary layers comprise three regions: boundaries near thesuction side, those near the pressure side, and the wake region behind the trail-ing edge. In most of the blade shapes the curvature of the suction side is fairlylow and the pressure and velocity gradients are not very large compared to theboundary-layer thickness. This kind of turbulent boundary layer can be simulatedusing zero-equation turbulence models just as well as with a more complicated andless efficient one- or two-equation models of turbulence. For the wake region nearthe trailing edge, there are no widely used and successful models to simulate thiscomplex flow. The Baldwin–Lomax model [32] is very compatible and has severalgood features, such as its usefulness in separated flows with small separation, itsrelatively smooth continuous transition of the length scale of the boundary layer intothe length scale of the far wake, and its ability to accurately predict the wall effectnear the trailing edge. For this reason, the Baldwin–Lomax turbulence model wasemployed in this study to handle the turbulent flow around the turbine cascade flows.

3.5 Decomposition of Flux Vector

3.5.1 Governing equations

In general, there are two ways to split the flux. One is the faster convergent schemecalled the Flux Vector Splitting (FVS), and the other is a more accurate schemecalled the Flux Difference Splitting (FDS). Of these, FDS has been shown tobe the more accurate one. One-fifth to one-half as many points were required toresolve a shear layer using fluxes in a first-order scheme compared to using otherfluxes in second- or even third-order schemes. However, in the FDS scheme, theexact Jacobian of the Roe matrix is expensive to compute, whereas approximatedJacobians have observed to give poor convergence compared to FVS.

The basic Roe scheme could produce entropy-violating solutions, and it has aslower convergence compared to FVS. In the present study, the FVS scheme is usedto evaluate the Jacobians in matrix equations. The flux vector can be decomposedinto two subvectors according to the positive and negative eigenvalues. Througha forward straight manipulation, one can compute the Jacobians A = ∂E/∂Q andB = ∂F/∂Q through the following expressions:

E = AQ (16)

F = BQ (17)

Equation (15) can be rewritten as

Qt + AQξ + BQη = Rξ + Sη (18)



Note that E and F are homogeneous functions of degree 1 in “Q.” The Jacobianshave the following transformation relations:

P−1AP = A (19)

P−1BP = B (20)

where A and B are the diagonal matrices and can been written as:

A = diag[u, u, u + aL(ξ), u − aL(ξ)](21)= diag[λE1, λE2, λE3, λE4]

B = diag[v, v, v+ aL(η), v− aL(η)](22)= diag[λF1, λF2, λF3, λF4]

where λ’s are the eigenvalues of flux vectors E and F , a denotes the speed of soundin the physical domain, and

u = u∂ξ/∂x + v∂ξ/∂y (23)

v = u∂η/∂x + v∂η/∂y (24)

L(ξ) =√ξ2

x + ξ2y (25)

L(η) =√η2

x + η2y (26)

According to the vector splitting method, the eigenvalues of the flux vector can beexpressed using positive and negative forms in order to split the flux vectors intotwo subvectors, that is,

E = (A+ + A−)Q = E+ + E− (27)

F = (B+ + B−)Q = F+ + F− (28)

The subvectors E+, E−, F+, and F− can be expressed according to the positiveand negative eigenvalues as:

E± = Jρ

2γ

⎡⎢⎢⎢⎢⎣2(γ − 1)λE1 + λE3 + λE4

[2(γ − 1)λE1 + λE3 + λE4]u + a(λE3 − λE4)ξx/L(ξ)[2(γ − 1)λE1 + λE3 + λE4]v+ a(λE3 − λE4)ξy/L(ξ)

[2(γ − 1)λE1 + λE3 + λE4](u2 + v2)/2 + a(λE3 − λE4)[uξx/L(ξ)+vξy/L(ξ)] + (λE3 + λE4)a2/(γ − 1)

⎤⎥⎥⎥⎥⎦(29)



F± = Jρ

2γ

⎡⎢⎢⎢⎢⎣2(γ − 1)λF1 + λF3 + λF4

[2(γ − 1)λF1 + λF3 + λF4]u + a(λF3 − λF4)ηx/L(η)[2(γ − 1)λF1 + λF3 + λF4]v+ a(λF3 − λF4)ηy/L(η)

[2(γ − 1)λF1 + λF3 + λF4](u2 + v2)/2 + a(λF3 − λF4)[uηx/L(η)+vηy/L(η)] + (λF3 + λF4)a2/(γ − 1)

⎤⎥⎥⎥⎥⎦(30)

The FVS form of equation (7) can be written as:

Qt + E+ξ + E−

ξ + F+η + F−

η = Rξ + Sη (31)

3.5.2 Special treatment of the artificial dissipation terms and numericalalgorithm

In this section, the new way to treat the artificial dissipation terms is proposed.According to the idea proposed by Turkel and Vatsa [15], the dissipation terms canbe written in the matrix-valued dissipation. However, in the present method, thedissipation term was not incorporated into the convection flux term as proposedby Turkel and Vatsa. The idea to overcome the effect of eigenvalue stiffness on theconvergence of a compressible flow calculation was incorporated into the artificialdissipation terms. The dissipation terms were incorporated into the time-derivativeterms. The dissipation terms can play a role in relatively enlarging the eigenvalue ofthe time-derivative term. The dissipation terms that required stabilizing the schemeis implemented in a convenient manner by modifying the convective fluxes asfollows:

Fξ+1/2,η = 0.5 ∗ (Fξη + Fξ+1,η) − Dξ+1/2,η (32)

Eξη+1/2 = 0.5 ∗ (Eξη + Eξη+1) − Dξ,η+1/2 (33)

The terms Dξ+1/2,η and Dξ+1/2,η represent the dissipative terms in ξ andη directions,respectively.Two forms of artificial dissipation models can be used. One is the scalardissipation model and another is the matrix-valued dissipation model.

The concept of the matrix-valued dissipation model was adapted and modifiedto form a new numerical scheme. The spectral radii for the ξ direction is:

λξ = |u| + aL(ξ) (34)

To form the present scheme and optimize the dissipation model in the sensethat the same dissipation scaling is used for all the governing equations in agiven coordinate direction, the matrix-valued dissipation term in ξ direction can bewritten as:

DE = E · ∂Q∂tt (35)



Following the matrix-valued dissipation concept, matrix in ξ direction can bewritten as:

F = λ3I +(λ1 + λ2

2− λ3

)×

(γ − 1

a2 E1 + E2

L(ξ)2

)(36)

+ λ1 − λ2

2aL(ξ)× [E3 + (γ − 1)E4]

where

E1 = [1, u, v, cpT ]T [q2/2, −u, −v, 1]

E2 = [0, ξx, ξy, u]T [−u, ξx, ξy, 0]

E3 = [1, u, v, cpT ]T [−u, ξx, ξy, 0]

E4 = [0, ξx, ξy, u]T [q2/2, −u, −v, 1] (37)

where q is the total velocity defined as:

q2 = u2 + v2 (38)

and

λ1 = u + aL(ξ)

λ2 = u − aL(ξ) (39)

λ3 = u

However, in the calculation, we cannot choose λ1, λ2, and λ3 according toequation (39) because near the stagnation points, λ3 approach zero, whereas nearsonic lines λ1 and λ2 approach zero. For solving this problem, we limit these valuesin the following manner:

λ1 = max (u + aL(ξ), u/M 2r )

λ2 = max (u − aL(ξ), u/M 2r ) (40)

λ3 = max (u, u/M 2r )

where

Mr =⎧⎨⎩

0.001M 2

1

M ≤ 0.0010.001 < M ≤ 1

M > 1(41)

and where M is the calculated Mach number.



The matrix E can be represented in the form ±E according to the positive or

negative value of λ1, λ2, and λ3, that is,

E = (+E −−

E )/2 (42)

Similar expression can be derived for DF by replacing the contravariant velocity uby v and ξ by η in equations (35)–(42). By adding the matrix form of the dissipativeterms in the governing equation (31), we obtain:[t

(∂+

A

2∂ξ− ∂−

A

2∂ξ+ ∂+

B

2∂η− ∂−

B

2∂η

)+ I

]Qt + E+

ξ + E−ξ + F+

η + F−η = Rξ+ Sη

(43)where I represents the unit matrix, and +

A ,−A ,+

B , and −B denote the matrices

corresponding to the eigenvalues of the convection flux matrices.By using the second-order time-difference algorithm, we have

Qn+1 − Qn

t= 1

2

[(∂Q

∂t

)n+1

+(∂Q

∂t

)n]

(44)

[t

(∂+

A

2∂ξ− ∂−

A

2∂ξ+ ∂+

B

2∂η− ∂−

B

2∂η

)+ I

]Qn+1 − Qn

t

= −(∂E+

∂ξ+ ∂E−

∂ξ+ ∂F+

∂η+ ∂F−

∂η

)n+1

+[∂R

∂ξ+ ∂S

∂η

]n

(45)

Recalling that (∂E

∂ξ

)n+1

=(∂E

∂ξ

)n

+ ∂[An

(Qn+1 − Qn

)]∂ξ

(46)

(∂F

∂η

)n+1

=(∂F

∂η

)n

+ ∂[Bn

(Qn+1 − Qn

)]∂η

(47)

Considering the stability, the forward and backward scheme were adopted in thedifference of flux splitting. We can define that:

Qn+1i,j = −t

[∇ξE+

ξ+ ξE−

ξ+ ∇ηF+

η+ ∇ηF+

η

]n

+t

(cR

ξ+ cS

η

)n

(48)

δQn+1i,j = Qn+1

i,j − Qni,j (49)

where the backward operator, for example in ξ for the E term, can be writtenas ∇ξE = 0.5 (3Eξ − 4Eξ−1 + Eξ−2), whereas the forward operator ξE = −0.5(3Eξ − 4Eξ+1 + Eξ+2) and ξ = 0.5 (Rξ+1 − Rξ−1).



By substituting equations (46) and (47) into equation (45) and considering thedefinition in equations (48) and (49), we can obtain the following difference scheme:

dW δQn+1i−1, j + dSδQ

n+1i, j−1 + dPδQ

n+1i, j + dN δQ

n+1i, j+1 + dEδQ

n+1i+1, j = Qn+1

i, j (50)

where

dW = − t

2ξ(+

A )ni−1,j − t

2ξ

(A+

i−1,j

)n

dS = − t

2η(+

B )ni,j−1 − t

2η

(B+

i,j−1

)n

dP = 1 + t

2ξ(+

A )ni,j − t

2ξ(−

A )ni,j + t

2η(+

B )ni,j − t

2η(−

B )ni,j

+ t

2ξ

(A+

i,j

)n − t

2ξ

(A−

i,j

)n + t

2η

(B+

i,j

)n − t

2η

(B+

i,j

)n

dN = t

2η(−

B )ni,j+1 + t

2η

(B−

i,j+1

)n

dE = t

2ξ(−

B )ni+1,j + t

4ξ

(A−

i+1,j

)n

In the above expressions, dW , dS , dP, and dE are scalars, which can be calculateddirectly from flux vectors. It is shown that the present scheme mainly includes bothimplicit and explicit parts. Equation (50) is the implicit part of the present scheme,which can improve the numerical stability when a large time step is used in thecalculation. Because of the application of flux vectors, the computational effortscan be greatly reduced as compared to other coefficient matrix implicit schemesfor any time step [2]. Equation (48) is the explicit part of the scheme. We foundthat the implicit part, equation (50), can be solved by using a two-sweep method,which is similar to the hopscotch-sweep method but along i (i.e., ξ) and j (i.e., η)directions. Equation (50) can then be written in the form of the two-sweep form asfollows:

dSδQn+1i,j−1 + dPδQ

n+1i,j + dN δQ

n+1i,j+1 = Qn+1

i,j − dW δQn+1i−1,j − dEδQ

ni+1,j (51)

Calculation starts from i =1 to imax along the i-direction. At each station i ofthe time step n + 1, the variables δQi,j−1, δQi,j , and δQi,j+1 can be obtained alongthe j-direction. It is clearly shown that the present scheme consists of an explicitpart of Qi,j at any time step “n” and an implicit δQi,j at time step “n + 1.”

3.6 Stability Analysis

Stability is one of the important aspects in developing a numerical scheme, andonly a stable scheme can be used to simulate the differential equations. In this



section, the stability analysis of the present numerical scheme is addressed. Oneof the most common methods of stability analysis is the von Neumann stabilityanalysis. With this method, a finite Fourier series expansion of the solution to amodel equation is made, and the decay or amplification of each model is separatelyconsidered to determine whether the method is stable or not. Analysis showedthat the typical stability limitation for the explicit scheme equation (49) is the CFLnumber restriction. Following the von Neumann analysis, it was shown that thepresent implicit scheme is unconditionally stable.

Nonetheless, the determination of stability condition for the present hybridscheme is quite difficult due to the complication of the equation systems employedfor the computation. To simplify the analysis, let all η derivatives in equation (47)equal to zero and all the eigenvalues be greater than zero so that in ξ-direction wehave |A−| = 0 and |A−| = |P+

E P−1|. In the stability analysis, only one of the twovelocity components was performed. The stability analysis of the other variablesis the same. The experience shows that, if each variable in the equation system isstable, the entire equation system in general is stable. Suppose that ρA and ρM are,respectively, the spectral radii of the matrices A and M , where

M = ∂R

∂(∂Q∂ξ

) (52)

The numerical scheme of equations (48) and (49) can be written in the followingscalar forms:

un+1i = −tξ

[∇ξE+

ξ

]n

+tξ

(cR

ξ

)n

(53)

(−tξρA

ξ

)(un+1

i−1 − uni−1) +

(1 + tξρA

ξ

)(un+1

i − uni ) = un+1

i (54)

Upon substitution of the Fourier component of the solution into equations (53) and(54), we have:

un+1i+1 = V n+1e(i+1)θ

√−1 (55)

The following amplification factor for equations (53) and (54) can be obtained:

|G|2 =ξ + tξ

ξρM [−2 + 2 cos (mξ)]

2 + [tξρA sin (mξ)

]2[ξ +tξρA −tξρA cos (mξ)

]2 + [tξρA sin (mξ)

]2 (56)

The stability condition of the scheme is that the absolute value of G is less than orequal to unity for all wave numbers m. From equation (56), the following stabilitycondition can be obtained:

tξ ≤ ξ

ρA + 2ρMξ

(57)



From a similar analysis, letting all ξ derivatives in equation (50) equal to zero andletting all the eigenvalues be greater than zero, we can obtain:

tη ≤ η

ρB + 2ρNη

(58)

where ρB and ρN represent the spectral radii of the matrices B and N , respectively,and N is defined as:

N = ∂S

∂(∂Q∂η

) (59)

It is clearly shown that the stability constraint of equations (57) and (58) is due tothe explicit part. Therefore, we can conclude that the present implicit scheme isunconditionally stable.

By solving the spectral radii of the flux Jacobian matrices for the convectiveterms and diffusion terms, we can obtain the following stability conditions:

tξ ≤ ξ

|u| + a√ξ2

x + η2y + 2ω

ξρ

(ξ2

x + η2y

) (60)

tη ≤ η

|v| + a√ξ2

x + η2y + 2ω

ηρ

(ξ2

x + η2y

) (61)

where

ω = max (µ, λ+ 2µ,µγ/Pr)

The stability analysis shows that the stability restraint arises from the explicit partfor the present scheme. The implicit part of the scheme is unconditionally stable.However, from the calculation point of view, the explicit part must be treatedwithin a certain time step. When the time step increases, the implicit scheme isactivated. From this point of view, the present overall scheme can be judged to beunconditionally stable. From the practical point of view, however, many factorssuch as smoothing of the implicit calculation, local linearization of the nonlinearequations, and complex geometry, all will constrict the time step. Therefore, thetime step must be selected according to the CFL number as:

t = min (tξ,tη) (62)

where

tξ ≤ CFL ·ξ|u| + a

√ξ2

x + η2y + 2ω

ξρ

(ξ2

x + η2y

) (63)



tη ≤ CFL ·η|v| + a

√ξ2

x + η2y + 2ω

ηρ

(ξ2

x + η2y

) (64)

From equations (63) and (64), it is easily found that when CFL ≤ 1, the time stepconstriction is the restriction of the typical explicit scheme. For the present schemethe CFL number can be chosen to be around 15. Anyhow it is clear that the presentscheme is more economical than a pure explicit scheme.

3.7 Applications in Turbine Cascade

In a practical use of the code, the convergence criterion is one of the importantfactors of the CFD program.There are several different kinds of convergence criteriato check the convergent state of the numerical solutions. In this work, two convergentcriteria are used to check the convergence of the calculation. One is the RMS residueof the four independent variables and the other is the mass flux error. They aredefined as follows:

Error(RMS) = 100 ×√(u)2 + (v)2 + (ρ)2 + (e)2

t(65)

Error(m′) = 100 × ∣∣m′0 − m′

EXIT

∣∣/m′EXIT (66)

The actual computational case studies are given in the following subsections.

3.7.1 C3X turbine cascade

The present numerical scheme was tested for a two-dimensional compressible flowover a C3X turbine cascade [2,34] by using the algebraic H-mesh as shown inFigure 3.2. The flow considered corresponds to the case 144 of the experimentconducted by Hylton et al. [34]. The geometric inlet and exit angles of the bladeare 0 degrees and 72 degrees, respectively. The flow conditions at the inlet are totalpressure PTo = 7.899 kPa, total temperature TTo = 815K, and the inlet Reynoldsnumber based on the true chord Re = 0.63 × 106. The ratio of exit static to inlettotal pressure is 0.59. The flow was assumed to be fully turbulent in the entirecomputation domain.

The effects of the mesh refinement study for case 144 were conducted prior to thecomputations. The different levels of the meshes based on the basic size 43 × 12 areshown in Figure 3.3. The study showed that the mass flux residual tends to approacha certain level with the increase in the mesh size. It was suggested that the meshsize of 86 × 24 is sufficient for a grid-independent condition. In order to assess theimpact of the present scheme, the study of the influence of the CFL number wasinvestigated. Figure 3.4 shows the typical convergent histories with different CFLnumbers (1, 5, 7, 9, and 11) by using a mesh distribution of 86 × 24. The researchshows that when the CFL number increases, the convergent rate is improved. Theresults also show that when the CFL number is greater than 7, the improvementin the convergent rate is small. If the CFL = 1, the computational performance is



(b) Body-fitted H-mesh(a) Algebraic H-mesh

Figure 3.2. Computational meshes.

2.01.5

Magnification factor for mesh size

1.00.0

0.1

0.2

0.3

0.4

|(m

in

mou

t)/m

out|%

0.5

0.6

0.7

2.5 3.0

Figure 3.3. Mesh refine study.

similar to the case of explicit scheme. In the present computation, the CFL numberis selected in the range between 5 to 11. It is clear that the present scheme is moreeffective than the explicit scheme. It is well known that refinement of the high-gridaspect ratio influences the convergence of the calculation.

Figure 3.5 shows good agreement of the present predictions for surface staticpressure distribution with the experimental data measured by Hylton et al. [34] andN–S computations by Kwon [35]. Figure 3.6 presents the computed Mach number



Time step

Err

or (

%)

010−4

10−3

10−2

10−0

10−1

200 400 600 800

CFL = 1

CFL = 5

CFL = 7CFL = 8CFL = 11

Figure 3.4. Computational convergent histories.

1.0

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.00.80.60.40.20.0

x/c

Present (pressure side)

Present (suction side)Kwon (pressure side)

Kwon ((suction side)Experiment

p/p 1

1

Figure 3.5. Blade surface static pressure distributions.



0.12

0.23 0.41

0.87

0.87

0.87

Figure 3.6. Contour of computed Mach.

contours. The results show that the flow is entirely subsonic and the computedmaximum Mach number is about 0.87. The wakes associated with the airfoils areclearly seen in the figure. It is also shown that the stagnation point occurs near thenose of the blade surface. These features are consistent with those expected in acascade flow.

Next, the heat transfer coefficient was computed and compared with theexperimental data. It can be seen in Figure 3.7 that the computed heat transfercoefficient is in good agreement with the experimental data on the pressure surfaceexcept near the leading edge. However, the computed heat transfer coefficient wasoverpredicated in the downstream region immediate vicinity of the leading edgealong the suction surface. This discrepancy may be caused by the heat transfer eval-uation through the energy equation that is coupled with the momentum equations.In the experiments, the flow field starts from laminar at the upper stream and thenundergoes transition to become fully turbulent. However, in the calculation, theflow was assumed to be fully turbulent resulting in the overprediction of the heattransfer coefficient near the leading edge of the suction surface.

Figure 3.8 shows the convergent history when using these two types ofH-meshes. There are slight differences appearing between the two mesh mod-els. It seems that the algebraic H-mesh has a better convergent property than thebody-fitted H-mesh.



1.0

0.9

0.8

0.7

0.6

0.5H/H

0.4

0.3

0.2

0.1

0.00.0 0.2 0.4 0.6


Experiment (suction side)

Experiment (pressure side)

Present (suction side)

x /c

0.8 1.0

Figure 3.7. Heat transfer coefficient distribution on blade surface.

Time step

Err

or (

%)

010−4

10−3

10−2

10−0

H-meshBody fitted mesh

10−1

200 400 600 800

Figure 3.8. Mesh influence of the convergent histories.



−72

−74

−76

−70

−80

−82

−84

5000 1000 1500Time step

Out

flow

ang

le

Present

Experiment

Denton

Figure 3.9. Convergent histories of the outflow angle.

3.7.2 VKI turbine cascade

Another test case of the present algorithm is the flow around theVKI turbine cascade.In the calculation a 77 × 21 H-type mesh, which is similar to that shown in Figure3.2a, was used with the same surface definition as suggested by Xu and Amano aswell as Kwon [2,35]. For comparison with other computational results, the meshpoints were also selected to be the same as those suggested by Sieverding [38].The flow conditions were the same as in his experiments. Figure 3.9 shows theconvergence history for the average outflow angle. The present computations showthe angle effectively reaching a steady value, which is in good agreement with theexperiment [37]. Further noted in the figure is the convergent rate of the presentscheme being slightly better than that in the method proposed by Denton [37]. Thecomputed outflow angle with different isentropic exit Mach numbers is shown inFigure 3.10. The isentropic Mach numbers were calculated using the Bernoulli’sequation with total pressure at the entrance and static pressure on the wall. Theresults show that the present calculation agrees well with the experiment. Thecalculated exit angle when the exit isentropic Mach number is 1.2 is almost the sameas that of Denton. Figure 3.11 shows the computed blade surface isentropic Machnumber distributions compared with both the experiment and computed results byDenton [37]. It shows that the present computation agrees with the experimental



0.6 0.7 0.8 0.9 0.1 1.1 1.2

−79.0

−79.5

−80.0

−80.5

−81.0

Denton

Experiment

Persent

MIEXIT

Out

flow

ang

le

Figure 3.10. Outflow angle with Mach number.

Pressure (Denton)

Suction (Denton)


Present (suction side)

Experiment

x/c

3.00.0

0.2

0.4

0.6

0.8

M10

1.0

1.2

3.2 3.4 3.6 3.8 4.0

Figure 3.11. Isentropic Mach number distributions on the blade surface.



10

10

0 4

0 7

0 8

Figure 3.12. Contour of computed Mach number.

ones when compared with those by Denton. Figure 3.12 is Mach number contourbetween the cascade blades.

An efficient numerical scheme for computations of the flow and heat transfer inturbine cascade flow was proposed and tested. It was demonstrated that flux-vectorscheme has the following advantages:

• The flux-vector scheme avoids the coefficient matrix operation and omits anyiteration within the same time-step calculation. Therefore, less computationaleffort is needed in the calculation.

• The present algorithm possesses the positive features of the explicit and implicitalgorithms. Because the present algorithm did not use approximation factor (AF)in the time-marching process, the CFL number can be chosen to be large enoughto save computational time. Furthermore, since the present scheme includes animplicit part, the method improves the convergence ability especially for thelarge time-step computations.

• The computation shows that the present scheme has a similar convergent ratecompared with the multigrid method developed by Denton [37]. The upwindscheme is used in the FVS scheme in order to improve the stability of the overallscheme.

• The flux-vector method has several merits such as good convergence and overallaccuracy in predicting the flow field and heat transfer around the turbine cascades.This fact was supported by the validation test shown in good agreement withexperiments.



In the present method, the dissipation terms were incorporated into the time-derivative terms. This character renders the scheme more accurate as well as avoidsthe eigenvalue stiffness when the calculation is performed in a low Mach numberrange and accelerates the convergence of the simulation. Finally, it is shown that thepresent scheme is more advantageous than common implicit and explicit schemes.The present scheme can be easily extended to full three-dimensional turbine flowcalculations.

3.8 Numerical Method for Three-Dimensional Flows

The Navier–Stokes equations can be written in a body-fitted curvilinear coordinatesystem (ξ, η, ζ) as follows:

∂(J −1U )∂t

+ ∂F

∂ξ+ ∂G

∂η+ ∂H

∂ζ= ∂FV

∂ξ+ ∂GV

∂η+ ∂HV

∂ζ(67)

where the convective terms can be written as:

∂F

∂ξ= A

∂U

∂ξ(68)

∂G

∂η= B

∂U

∂η(69)

∂H

∂ζ= C

∂U

∂ζ(70)

The detailed expressions in the above equations can be found in Turkel and Vatsa[15]. Following the previous section, a dissipation model employed in this study isa nonisotropic model, in which the dissipative terms are given as a function of thespectral radii of the Jacobian matrices associated with the appropriate coordinatedirections. The dissipative terms in the three directions of the coordinates haveforms similar to that discussed in previous section and in Turkel and Vatsa [15].This method can overcome the effect of eigenvalue stiffness on the convergenceof an incompressible flow calculation by using the time-matching scheme. Thedissipation terms were incorporated into the time-derivative terms. In this way, thedissipation terms can enlarge the eigenvalue of the time-derivative terms, which willbenefit the flow solver convergence, in particular when the code is used to solvethe low-speed flow problems. The dissipation term that is required to stabilize thescheme is implemented in a convenient manner by modifying the convective fluxesas follows:

Fξ+1/2,η,ζ = 0.5 ∗ (Fξ,η,ζ + Fξ+1,η,ζ) − dξ+1/2,η,ζ (71)

Gξ,η+1/2,ζ = 0.5 ∗ (Gξ,η,ζ + Gξ,η+1,ζ) − dξ,η+1/2,ζ (72)

Hξ,η,ζ+1/2 = 0.5 ∗ (Hξ,η,ζ + Hξ,η,ζ+1) − dξ,η,ζ+1/2 (73)



The terms dξ+1/2,η,ζ , dξ,η+ 1/2,ζ , and dξ,η,ζ+1/2 represent the dissipative terms in ξ−,η−, and ζ− directions, respectively, which can be found in Xu and Amano [1,3].Using equations (2)–(4), the time derivative of dissipative terms can be written intoa matrix form as:

D = [1, dξ+1/2,η,ζ , /u, dξ,η+1/2,ζ/v, dξ,η,ζ+1/2/w, 1]T (74)

By substituting the dissipative terms of equation (74) into equations (71) through(73) and incorporating into equation (67), we obtain:(

I + D

U

)∂(J −1U )

∂t+ ∂F

∂ξ+ ∂G

∂η+ ∂H

∂ζ= ∂FV

∂ξ+ ∂GV

∂η+ ∂HV

∂ζ(75)

The second-order numerical dissipation terms were set so that the variables decreaseexponentially as a function of time. This technique allows a computation to dampout high-frequency errors at the initial stage of iterations and, therefore, avoidsthe second-order numerical dissipation terms to dissipate when the computationapproaches a converged state. A finite-volume algorithm based on a Runge–Kuttatime-stepping scheme [15] is used to obtain the steady-state solutions by solvingthe Navier–Stokes equations. The turbulence model used in this study is a Baldwin–Lomax turbulence model [32]. The meshes generated in this study are hexagonalmeshes; the grid lines near the solid wall are almost normal or parallel to the solidwalls. In order to simplify the programming effort, the turbulence equations wereapplied along grid lines rather than in the direction normal to the solid boundaries.This avoids flow calculations in all the normal directions to the solid wall andthe necessary interpolations of the flow variables. During the calculations, it isdifficult to keep all the y+ values within a small level, especially close to the cornerregion. Therefore, the wall function was used in the calculation. The static pressurecoefficient is defined as:

Cps = P − P01

p01 − ps2(76)

The total pressure loss is defined as:

Cp0 = P01 − P0

p01 − ps2(77)

where ps2 represents the mass average static pressure of the blade exit plane.

3.9 Applications of Three-Dimensional Method

3.9.1 Analysis of pitch-width effects on the secondaryflows of turbine blades

The inlet and outlet boundary conditions used in the computations were takenfrom measurement. The total pressure at inlet was equal to the test [38] for all the



calculations. At the exit plane, one of the most critical conditions was to specifythe static pressure. The average exit pressure in the experiment [38] was used in thecalculations. The asymptotic boundary conditions with second-order streamwisederivatives of variables were set to zero. The overall mass conservation through theblade flow passage is imposed by correcting the outlet velocity components with thecalculated velocity profiles on the plane next to the outlet and inlet flow mass duringiterations. The no-slip wall boundary conditions are used for all the solid walls.

Grid-converged solutions for the high-speed viscous flows over a turbine bladeshould be obtained with sufficiently fine meshes. However, a computation with finemeshes is very time consuming and difficult to assess the numerical accuracy ofthe solution. In fact Inoue and Furukawa [23] reported that it was difficult to opti-mize the artificial dissipation coefficients by using a common method that impairedaccuracy and reliability of the schemes to predict a cascade performance. To pre-dict a high Reynolds number turbulent flow, highly clustered grids are requiredtoward the wall. The grid refinement study was conducted through the computa-tional uncertainty study. The relationship between the calculated mass residual,100 × |min − mout|/mout %, and various grid sizes was tested to identify the pointwhere the mass residual reaches its asymptotic value. In this study, the systematicgrid refinement studies are performed. The grid refinement study was conducted byusing Richardson extrapolation method [39]. The mass factor, f = |1 − min/mout|,is used to evaluate the grid refine study. Using the fine grid calculation with amagnification factor of 3, we obtain f1 = 0.18 × 10−4. Then we change the gridwith the magnification factor to 2, and obtain f2 = 0.85 × 10−4. In this case, theerror is ε= 100 × (f2 − f1)/f1 = 0.67%. For a grid increase rate, IN = 1.5, the finegrid value of the grid convergence index (GCI) for the present second order isGCI = 3ε/(IN2 − 1) = 1.61%. Although the confidence in the GCI as error band isnot justifiable, it shows the current grid structure to be conservative. It also showsthat the uncertainty in the current calculation for the mass conservation is within1.61%. The distance of the first mesh from the solid surface was chosen so that amaximum y+ value becomes less than 15, which is considered to be fine enough forthe present calculations. The mesh has 45 × 41 × 115 node points in the pitchwise,spanwise, and streamwise directions, respectively, which was identified to an opti-mal situation as a grid-independent state. The blade-to-blade computational mesh isshown in Figure 3.13. The mesh is formed consisting of three zones: upstream of theblade, inside the blade, and downstream of the blade sections. For comparison withthe test results [38], the outlet of the calculation plane was selected to be perpendic-ular to the x-direction (axial direction), which corresponds to the tested exit plane.

Most CFD studies [40–42] on turbomachines were mainly based on a linearcascade of high-turning turbine blades where the secondary flow was very strong.In this study, an annular turbine blade was focused, which is similar to actual turbinestator vanes. The airfoil model investigated by Sieverding [38], where the blade testswere cited from NACA TN D-3751, is used for validation of the code. The tip androot radii are 0.355 and 0.283 m, respectively. The stagger angle is 42.5 degrees withrespect to axial direction and the blade axial width length, Ca = 0.087 m. The bladeprofile is constant along the blade height and is untwisted. Three root pitch-to-width



(a) Blade-to-blade mesh at middle span (45 x 117) (b) Meridional mesh (41 x 117)

Figure 3.13. Calculation meshes.

0

10−1

10−2

100

Mas

s er

ror

No. of steps

1000 2000 3000 4000 5000 6000 7000

Figure 3.14. Convergence history.

ratios, b/Ca = 0.639, 0.973, and 1.460, are studied here. The root pitch-to-widthratio, b/Ca, was varied by altering the number of blades. Three blade numbers wereused in the calculation: 26, 21, and 14. These calculations were compared with theexperimental data of the annular turbine blades [38].

The computations were assumed to be converged when the mass flow errorbecame less than 0.1% of the inlet mass flow rate. The mesh used in this studyand typical convergence history are shown in Figure 3.14. The convergence speedbecame slow after the mass flow error became close to 0.1%. In all the plots shownin this section, the viewpoint is set to look in the downstream direction from the



0 10 20 30 40 50 60 70

Bet

a

−40

−50

−60

−70

−80

−90

Blade Height

Figure 3.15. Average spanwise flow angle distribution at x/Ca = 0.9 (dot is exper-imental results).

(a) b/Ca = 0.639 (left is leadingedge and top is casing).

(b) b/Ca = 0.986. (c) b/Ca = 1.460.

Figure 3.16. Static pressure coefficient distributions on the suction surface of theblade.

upstream location, with the pressure surface on the left-hand side and the suctionsurface on the right-hand side.

Figure 3.15 shows the comparison of the experimental results [38] with thepresent computations for the average spanwise flow angle distribution located atx/Ca = 0.9. The computations are in agreement with the experiment. Figure 3.16shows the static pressure coefficient distributions near the suction surface of bladesfor different root pitch-to-width rations, b/Ca. The inception and transport of thepassage vortex and its impact on the surface boundary layers present different



b /Ca = 0.639 b /Ca = 0.973 b /Ca = 1.460

Sa1Sa2

Sa2Sa1

Figure 3.17. Velocity vectors at the leading edge of the blade at 0.28% height fromroot.

characteristics for different b/Ca. There is a region near the wall corner where thechordwise and spanwise pressure gradients are small. It is also clear from Figure3.16 that the possible boundary layer separation lines occur for cases b/Ca = 0.639and 0.986, and they are located at the corners of the hub and trailing edge. Theseparation region for the case b/Ca = 1.460 is located at 50% of Ca position. Theseparation location seems to move forward as b/Ca increases. This feature confirmsthat the inception of the passage vortex developed near the airfoil leading edgeand grew fast when the pitch b/Ca was increased. Figure 3.16 also shows that thepressure coefficient on the suction surface is lower for larger b/Ca. This is because,as b/Ca increases, the flow constraint from the blade becomes small, and therefore,the blade’s local curvature plays an important role. For the case of b/Ca = 0.639and 0.986, a vortex develops faster near the blade tip and moves toward the casingbecause the local pitch size at the tip is much larger than that at the hub. In the caseof b/Ca = 1.460, the casing and the hub vortex have similar characteristics becauseb/Ca is large enough, thus causing the blockage to be small enough so that it doesnot restrict the flow. The flow characteristics are dominated by local curvature forb/Ca larger than 1.46.

The experimental data [43,44] indicated that one of the most important featuresin a turbine blade boundary layer is the occurrence of the separation and saddlepoint on the end wall near the leading edge. The predicted boundary layer flowpatterns for different b/Ca near the hub leading edge at 0.28% height are shownin Figure 3.17. The calculations show that there is a saddle point for all b/Ca.Two attachment lines, Sa1 and Sa2, and two separation lines, Ss1 and Ss2, can beidentified from the velocity vectors. The separation lines correspond to the two legsof the horseshoe vortices formed near the leading edge. The separation line Ss2is pushed away from the leading edge. The first separation line Ss1 moves aroundthe leading edge and moves against the freestream flow direction and it mergeswith the flow coming from the upstream direction to form a suction-side horseshoevortex. The separation line Ss1 is also developed along the suction side of the blade,which can be seen from the static pressure coefficient distribution in Figure 3.17.



b /Ca = 0.639 b /Ca = 0.973 b /Ca = 1.460

Figure 3.18. Velocity vectors in the meridional surface before the leading edge.

0.7 0.8

PS

0.9 1.0

1.1

1.2

1.3

13

HUB 12

b/Ca = 0.986 (experiment)

−0.85

−0.8−1.1

−0.95

−1.15

−0.75

−1.3

−1.25

−1.05

−0.9

−1.05

−1.2

−0.1

b/Ca = 0.986

−0.85

−0.8−0.7

−0.75−0.95

−0.9

−0.1−1.05−1.35

−1.55

−1.45

−1.4−1.5

−1.6

−1.15−1.1

−1.2 −1.15

−1.2−1.05

−1.1

−1.2−1.7

−1.35

−0.95

−0.9−1.25

b/Ca = 1.460

−0.8

−0.7

−0.6−0.5

−0.9

−1.0

b/Ca = 0.639

−1.1 −1.2

Figure 3.19. Static pressure coefficient contour at x/Ca = 0.9.

The locations of the saddle points are closer to the leading edge for larger pitchsizes. This is why some of the flow visualization [42] did not find a saddle pointif the experiment did not extend sufficiently far upstream. The velocity vectors inthe meridional plane upstream of the leading edge are shown in Figure 3.18. Thevector distributions show that there is a separation region occurring upstream ofthe leading edge. It is shown that the region of the separation depends on the pitch–width ratio, b/Ca. A lower b/Ca generates a relatively larger separation region. Thischaracteristic determines the location of the saddle points on the blade-to-bladeplane depending on the pitch–width ratio b/Ca. The present computations showthat there is a saddle point upstream of the leading edge attributed to the blockageof the blades. With increasing blockage, the secondary saddle vortices move moreforward and the separation in the front of the blade becomes smaller. The locationof the saddle point depends on the b/Ca as well as the local curvature of the blade.

Figures 3.19, 20, and 21 show the contours of the static pressure coefficientCps distributions at 90%, 100%, and 110% chord positions, respectively, fromthe leading edge for different values of b/Ca. The measurement results for –Cpsat 90% chord position is plotted for comparison with the calculations. It shouldbe pointed out that the measurement plane in the experiments [44] is defined to



−0.80−0.7

−0.65

−1.0

−1

−1.1 −0.95−0.85

−0.85

−0.8

−1.15 −1.05

−1.20

−1.05

−1

−1.1

−1.15

−0.85 −0.9−0.9565−0.7

−0.75

−1.35 −1.30.4

−1.25

−1.05

−0.95

−0.85

−0.8

−0.75−0.7

−0.65

−0.75

−0.8

−0.95

−0.85

−0.9

−1.1−1.35 −1.4 −1.3

−1.25

−1.05

−1.15

−1.85

−0.9−0.7

−0.65−0.8

−0.95

−1

−.−8

−1.0

−1.2

b/Ca = 0.639(contour value from−1.15 to −0.6 withincrement 0.05)

b/Ca = 0.973(contour value from

−1.3 to −0.6 withincrement 0.05)

b/Ca = 1.460(contour value from

−1.6 to −0.6 withincrement 0.05)

−0.75 −0.7

−0.65

Figure 3.20. Static pressure coefficient contour at x/Ca = 1.

−1.1 − 1.35 −1.3−1.25

−1.15

−1.05

−1.9

−0.85

−0.8−0.95

−0.9−0.85−0.8

−0.70−0.7

−0.75

−1

−1.05−1.95

− 0.85

−0.9

−0.8−0.75

−0.65−0.7−0.65

−1.1−1.45

b/Ca = 0.639(contour value from −1.1

to −0.65 with increment 0.05)

b/Ca = 0.973 (contour value from −1.35


b/Ca = 1.460 (contour value from −1.35


−1.3−1.25

−1.2

−1.05−1.45

−1.1−1.5

−1.95

−0.9

−0.85−0.8

−0.95

−0.950.9 −1

−0.85−0.80.5

−1.35

Figure 3.21. Static pressure coefficient contour at x/Ca = 1.1.

be perpendicular to the axial direction while the computations are along the meshdirection, which causes a slight discrepancy. Figure 3.19 shows that the compu-tation and measurement are in good agreement with each other. The results fordifferent b/Ca show that for smaller b/Ca, the lowest pressure becomes higher thanthose with larger pitch cases. The core of the secondary vortex occurs at differentlocations for all three cases. For the smaller b/Ca, the secondary vortex appearscloser to the corner of the hub on the suction side. For larger b/Ca, the vortex coremoves away from the suction-side toward the pressure side. When compared to thedifferent axial location of Cps, it is shown that the vortex core is developed andmoves away from the suction side, to the middle point between the suction andpressure-side surfaces. As the fluid moves out from the blade passage, the vor-tices become very small. This is because the blade-to-blade pressure gradient isreduced. This vortex core is part of the leading edge vortex. Initially, the vortexoccurs very close to the suction-side wall and then it develops toward the exit plane.At the same time, the horseshoe vortex at the pressure-side leg, Ss2, mixes withthe secondary flow and rapidly decays. This phenomenon is in good agreementwith the measurements for the pitch–width influence in the linear cascade [40].



b/Ca = 0.639 b/Ca = 1.460b/Ca = 0.973b/Ca = 0.973(experiment)

TIF

Figure 3.22. Total pressure loss contour at x/Ca = 0.9.

0.55

0.60

b/Ca = 0.639 b/Ca = 0.973 b/Ca = 1.460

0.60 0.80

0.75

0.75

0.70

0.65

0.55


This figure also shows that the corner vortex at the tip of the blade is not very strongand cannot be clearly observed in the computations.

The total pressure losses Cpo in Figures. 19, 20, and 21 further confirm theaforementioned conclusions that corner vortices occur on both the hub andthe casing suction-side corners. The computations for the case of b/Ca = 0.973are compared with the experiment [44] and are shown in Figure 3.19. The mea-sured Cpo values of the first line in the experiment are the same as the computations.It is shown that the computations are in fair agreement with the experiments, exceptat tip and hub regions where the computations show a larger vortex region than thatshown by the experiment.

The influence of the losses due to the pitch–width variation can be seen bycomparing Figures 3.22, 23, and 24 for different b/Ca. For the cases of b/Ca = 0.973and 1.46, the high-loss regions from both tip and hub corners near the suction sideare developed. These losses are caused due to the corner vortices and the suction-side pressure characteristics. The end wall boundary layer is very thin near thepressure side within the blade passage. The pressure-side boundary layer and thesuction-side boundary layer meet after the flow passes the blade trailing edge, asshown in Figure 3. 20. It is also shown that the total pressure losses Cpo increaseswith the increase in b/Ca. The loss contour shows that the secondary flow is strongerfor larger b/Ca, and it develops faster. This is because for large b/Ca, the flow controlcapacity become worse than that for small b/Ca. The secondary vortex is easy to



b/Ca = 0.639 b/Ca = 0.973 b/Ca = 1.460


b/Ca = 0.639 b/Ca = 1.460b/Ca = 0.973

0.02

0.01

0.01

0.01

0.11

0.11

0.12

0.12

0.01

0.03

0.02

0.08

0.08

0.08

0.080.08

0.02

0.08

0.06

0.05

0.050.03

0.07

0.07

0.06

0.06

0.06

0.06

0.01

0.020.02

0.02

0.05

0.010.01 0.03 0.03

0.08

0.08

0.09

0.09

0.09

0.04

0.1

0.1

+4

0.2

0.050.07

0.07

0.04

0.04

Figure 3.25. Blade-to-blade Mach number contour at 0.28% height.

develop in the blade channel. For the smallest b/Ca, the two passage vortices occurin the upper and lower parts of the blade passage closer to the suction side of theblade without strong interactions. With the increase in the b/Ca, the two vorticesstart interacting with each other. Owing to the occurrence of the passage vortices,and their interactions for the large b/Ca cases, the secondary flow is very strong.The strong three-dimensional effects dominate the whole passage flow for largeb/Ca. However, for smaller b/Ca, it has been found that the flow configuration isdifferent from those for larger b/Ca cases. The two vortices occur from lower andupper parts of the suction side, and they are more confined in the end wall region,and there remains a two-dimensional flow pattern for a wide blade height range.

The blade-to-blade Mach number contours for the hub boundary layer regionand the pitch and tip boundary layer regions are shown in Figures 3.25, 26, and 27,respectively, where the Mach number contours present different characteristics forthe boundary region and the main flow region. In the near-hub boundary region, asmaller b/Ca does not show a high Mach number region near the suction surface.It seems that the flow accelerates smoothly along the suction side for smaller b/Ca.For the cases of b/Ca = 0.973 and 1.460, a small region of a high Mach numberzone is observed close to the middle of the width in the suction side of the blade.It is shown that a separation zone occurs and expands with the increase in b/Ca.



b/Ca = 0.639 b/Ca = 0.973 b/Ca = 1.460

0.9 0.1 0.04

0.070.12

0.11

0.05

0.08

0.11

0.1 0.09

0.09

0.1 0.05

0.05

0.03

0.070.03

0.04

0.05

0.01

0.02

0.030.04

0.060.02

0.01

0.070.05

0.11

0.1

0.09

0.020.080.05

0.8

Figure 3.26. Blade-to-blade Mach number contour at 50% height.

b/Ca = 0.639 b/Ca = 0.973 b/Ca = 1.460

0.020.03

0.50.2

0.06

0.3

0.4

0.7

0.11

0.240.13

0.11

0.12

0.040.1

0.090.09

0.090.09

0.09

0.09

0.09

0.09

0.7

0.080.060.1 0.1

0.02

0.05

0.02

0.6

0.7

0.05

0.03

0.03

0.09

0.09

0.09

0.09

0.1

0.03

0.11

0.2

0.20.1

0.8

0.2

0.1

0.11

0.2

0.1

0.080.090.1

12

Figure 3.27. Blade-to-blade Mach number contour at 99.72% height.

The results show that the blade-to-blade Mach number gradient along the turbineaxial direction increases with the increase in b/Ca.

The blade-to-blade static pressure coefficient contours near the hub, pitch, andtip are shown in Figures 3.28, 29, and 30, respectively. It is shown that, due tothe strong secondary-flow effect, the minimum pressure occurs in the passage awayfrom the blade surface. Also note that, with the increase in b/Ca, the flow acceleratesfaster in the leading edge region. A clear low-pressure zone occurs on the suctionside of the blade for all the different pitch–width cases when a flow separation zoneon the blade suction surface.

Figures 3.31, 32, and 33 show the contours of the blade-to-blade total pressurecoefficient distributions in the hub boundary (0.28% height), pitch (50% height),and tip boundary (99.72% height) regions, respectively. The losses for differentb/Ca, the plots start with the same level at the same blade location. The total pressurelosses are much larger in hub and tip boundary regions than in the midspan for allcases. In general, small b/Ca has small losses for all flow regions. For the smallerb/Ca, a small zone of high loss extending from trailing edge can be observed only



b/Ca = 0.639 b/Ca = 0.973 b/Ca = 1.460

Figure 3.28. Blade-to-blade static pressure coefficient contour at 0.28% height.

−0.8

−0.8−0.8

−0.8−0.8

−1

−0.8−0.8

−0.1

−1

−1.1

−0.9−0.8−0.7

−0.6−0.5−0.2

−0.3−0.4−0.7

−0.7

−0.7

−0.6

−0.6

−0.5

−0.5

−0.4

−0.4

−0.2−0.3

−0.2

−0.1−0.1

−1−0.1

−0.9

−0.9−0.9

−0.9−0.9

−0.9−0.9−0.9

−1.4

−1.5−1.4

−1.7

−1.5

−11−11

−1.1

−1.7

−1.4−1.2−1.3

−1.1−1.1

−0.1−0.1

−0.3

−0.4

−0.7−0.6−1

−1

−0.09

−0.9−1

−0.8−0.08

−0.5

−0.3 −1.8

−1.4

−1.3

−1.3

−1.1−0.1−1.2

−1.2

b/Ca = 0.639 b/Ca = 0.973 b/Ca = 1.460

Figure 3.29. Blade-to-blade static pressure coefficient contour at 50% height.

b/Ca = 0.973

−+1

−0.1

−0.2

−0.3 −0.4−0.5

−0.6−0.8 −0.7 −0.7

−0.9−0.8

−0.9

−0.8−0.8−+0.7

−1

−1.1

−1.1

−1

b/Ca = 0.639

−0.4−0.1−0.6

−0.5

−0.6−0.6

−0.7−0.8

−0.7−0.6

−0.6−0.6−0.6

−0.7−0.8

−0.7−0.6

−0.6

−0.2−0.1 −0.5 −0.4

−0.3

b/Ca = 1.46

−0.3

−0.1

−0.2

−0.3−0.5

−0.1−0.8

−0.1

−0.6

−0.8−0.7

−0.9

−0.8

−+4 −1.1−1.9

−1.8−1.6

−1.5−1.7

−1.1

−1 −1−1

−1.3−1.5−1.1−1.6

−1.9−1.9

Figure 3.30. Blade-to-blade static pressure coefficient contour at 97.72 height.



1.4

1.41.31.2

1.5

1.5

1.5

1.51.31.5

1+7

1.41.3

1.41.5

1.4

1.7

1.61.1

1.31.6

1.61.6

0.9

0.7

0.5

0.4

0.50.6

0.90.3

.9

1.3

1.4

1.3 1.3

1.3

b/Ca = 0.639

0.8

0.9

1.11.2

1.5

1.51.51

1.51.71.5

1.5

1.3

1.1

1.5

1.6

1.6

1.51.51.5

1.51.7

1.8

1.61.9

1.7

1.7

0.7

0.7

0.7

1.3

1.5

1.51.51.51.51.5 1.4

1

b/Ca = 0.973

0.8

2.22.3

2.42.5

2221.5

1.8

1.6

1.6

1.7

1.7

1.7

1.4

1.4

1.4 1

2.42.5

2.32.2

2.22.52.42.71.5 1.1

1.3

1.3

1.51.51.6

1.41.5

1.7

1.9

0.9

0.9

1.8

1.4

1.1

1.1

b/Ca = 1.460

Figure 3.31. Blade-to-blade total pressure loss at 0.28% height.

1.41.5

1.5

1.6

1.1 1.4

1.2

1.2

1.1

1.1

1.6

1.2

1.3

1.3 1.3

1.1

0.80.9

0.8

0.90.9

0.6

1

0.9

0.60.9

0.60.90.80.7

1

1.2

1+8

111

1

1

11

1

1.1

0.60.6

0.6

+

111

0.8

0.8

0.8

0.80.7

0.9

1+120.4

.4322

9 31

11 1

12

9

1

211

122

2233

1

1

1

0.8

0.80.2

2.23.5

b /Ca = 0.639 b /Ca = 0.973 b /Ca = 1.460

Figure 3.32. Blade-to-blade total pressure loss at 50% height.

b/Ca = 0.639 b/Ca = 0.973 b/Ca = 1.460

1.71.1

1.12.21.3

1.9

1.6

1.8

1.3

1.21.11.1

1

1.32.21.91.7

2.4

1.21.3

2.42.42.4

1.3 1.3

1.3

1.3.8

0.8

0.7

1.8 1.1

1.3

1.4 1.7

1.6

1.11.31.6

1.7

1.4

0.1

1.2

123

1.7

1.21.2

1.2

1.10.7

1.21.3

0.6

1.80.6

0.7 0.8

0.8

0.9

0.90.80.91.31.21.1

1.1

Figure 3.33. Blade-to-blade total pressure loss at 99.72% height.



on the tip boundary. For a larger b/Ca, there appears a high-loss zone starting fromthe trailing edge extending to the upstream of the suction side of the blade.

3.9.2 Flow around centrifuge compressors scroll tongue

Centrifugal compressors utilize radial change across the impeller to raise gas com-pressor. A centrifugal compressor allows more diffusion within the impeller bladesto raise pressure; thus, centrifugal compressor can produce higher specific worktransfer within one stage compared to an axial compressor. Although the efficiencyof centrifugal compressor is lower than that of axial compressor at large volumetricflow rate, centrifugal compressor has many applications for moderate- and small-flow gas compressions. A centrifugal compressor has a shorter axial length withwider operating range than axial compressors to get the same discharge pressure.An increased efficiency of centrifugal compressors has been the traditional empha-sis of the compressor design. However, an increasing operating range and part loadefficiency have become more important. Many efforts have been made in researchfield to optimize different component of the compressor stage [45–51].

An industrial centrifugal compressor stage consists of three main components:a rotating impeller, vaned or vaneless diffusers, and a scroll or collector as shownin Figure 3.34. The impeller adds work to gas by increasing its angular momentum.Gas static pressure and velocity increase after passing through the impeller. Vanedand vaneless diffusers convert kinetic energy into static pressure by decelerating thefluid. A scroll is used for collecting gas from diffuser and delivering it to applicationsystem or next stage. The scroll is the last component in the compression process ina single-stage compressor. Gas leaves impeller with logarithmic spiral shape due tothe rotation of the impeller. A scroll has two functions: to collect and transport thefluid into downstream system and to raise the static pressure by converting kineticenergy into potential energy (static pressure). A scroll design is complicated dueto the effect of the strong three-dimensional swirl flow. Typically, a scroll design

Diffuser

Impeller

Scroll

Figure 3.34. Centrifugal compressor.



Sharp tongue

Leading edge Scroll discharge

Leading edge

Scroll

(a) (b)

Figure 3.35. (a) Sharp tongue. (b) Scroll tongue structures.

is based on the two-dimensional analysis. Traditionally the emphasis on the scrolldesign is mainly on collection and less on diffusion function. Diffusion inside ofscroll is more effective for the kinetic energy associated with the tangential velocity;this is accomplished by increasing cross-sectional area from scroll tongue to exit,creating a conical diffuser. But there is an efficiency loss caused by the inability ofthe scroll to convert portion of the kinetic energy due to the radial component offluid velocity out of the diffuser. This effect is compounded when a vaned diffuseris used, as vanes create a more radial flow at the vaned diffuser exit. Typically,there is a tradeoff between the increased recovery of vaned diffuser and the reducedrecovery of scroll; a different solidity vaned diffuser is used to meet performanceand operating range requirements.

At a scroll tongue as shown in Figure 3.35, the flow encounters a dividing plane,so-called tongue, above which the flow exits the compressor and below which theflow reenters the scroll. The tongue of a scroll is located at a 360 degree point, corre-sponding to the joint of scroll smallest area, largest area, and exit cone. The leadingedge of the scroll tongue is analogous to that of an airfoil; a sharp leading edge hasa less blockage effect but is sensitive to the angle of attack as shown in Figure 3.35a,while a well-rounded leading edge is insensitive to the flow incidence but has a largerblockage effect as shown in Figure 3.35b. A sharp tongue is very sensitive to theflow incidence, and causes a flow separation at off-design flows. The manufacturingprocess is very difficult for the construction of a sharp tongue. A rounded tongue,however, has a larger blockage area than a sharp tongue, which may reduce effi-ciency at design point. With the understanding of the influence of the tongue, a com-pressor with higher efficiency and wider stable operating region can be designed.For obtaining a better off-design performance, the research on a scroll tonguewas designed with a large cutback angle as shown in Figure 3.35b, which allowsrecirculation of about 25% of the stage discharge mass flow inside of the scroll.

Little research has been done to understand flows inside of a scroll, especiallyin the tongue area [52–54]. Recent research [55–59] found that a good scroll designcould improve the compressor performance and the operating range. Most scroll



Figure 3.36. Calculation meshes.

studies have been performed on existing geometries and can be categorized intotwo groups: scroll flow structure investigations [52–55] and flow interaction studiesbetween the scroll and the impeller [55–59]. Large portions of the scroll studieswere performed for pumps. There is little information available for the scroll designand the scroll parametric influences on the compressor performance. Recent pumpscroll studies [59] showed that the tongue shape impacts head, pressure fluctuations,and noise in a centrifugal pump. Irabu et al. [60] studied the water flow patternsaround a pump tongue at small flows.The study found that flow separations occurredfrom both upper and lower surfaces of the tongue, and further reported the flowfield influence due to different tongue geometries and focused on the detailed flowanalysis besides performance test on a large cutback tongue scroll.

Turbulence modeling and turbulence model selections are critical to the accuracyof the flow field simulations. A larger number of CFD codes for a turbomachineryanalysis use the zero equation turbulence model [1,2]. Most calculations showed thatzero equation turbulence models gave reasonable results for flow calculations. Someresearchers have demonstrated that other turbulence models can also be used to gainreasonable results [61,62]. Some comparisons for different turbulence models havebeen presented [63]. Different turbulence models showed different advantages. Forthe flow inside of the centrifugal compressor and scroll, different turbulence modelsalso had been used for flow calculations. There are very limited publications thataddress the comparisons of different turbulence models.

Two types of tongues, a sharp tongue (r/b2 = 0) and arounded tongue(r/b2 = 0.3), were investigated. The calculation meshes are shown in Figure 3.36.The CFD studies for both tongue shapes were performed by using the zero-equationturbulence model for comparisons of the tongue shape impacts. The CFD analysiswas also performed for rounded shape tongue volute by using the k-ε turbulencemodel to investigate the model effects of the turbulence equations in the CFD simu-lations. In the current study, six flow rates were calculated for volutes with differenttongues to provide the performance information. The computed performances fora round tongue with two different turbulence models are shown in Figure 3.37.



φ/φ0

ψ,η

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.6 0.8 1 1.2 1.4

AlgebraicAlgebraicExperiment

Experiment

K-EK-E

Figure 3.37. Comparison of the performance curves.

The calculations using the different turbulence models did not produce significantdifferences in the results of performance. The results from both two turbulencemodels are in good agreement with the experimental data. The calculations showthat only at the flow coefficient near choke, more choke margin occurred thanthe experimental data. The calculations also show that the two-equation turbu-lence models give slightly higher efficiency at design flow point compared with thezero-equation turbulence model. Furthermore, the k-ε turbulence equations pre-dicted a lower head coefficient at a larger flow and a lower flow compared with thezero-equation turbulence model.

However, the rounded tongue has less advantage over the sharp tongue whena compressor operates at a design flow rate. This is because when a compressoroperates at around the design flow condition, the flow near the sharp tongue areahas little incidence and the flow passes the sharp tongue smoothly. Therefore, thesharp tongue produces small blockage, whereas the rounded tongue produces largerblockage when the compressor operates near the design flow condition. As a result,at the design flow condition, the rounded tongue has less pressure recovery whencompared with the sharp tongue case. However, when a compressor operates ata flow rate larger than the design flow rate, a scroll tongue encounters a positiveincidence. Because a rounded tongue is not sensitive to the flow incidence, inci-dence losses for a rounded tongue are smaller than those for a sharp tongue. Therounded tongue produces better pressure recovery than the sharp tongue. When thecompressor operates at a low flow rate, the advantages of the sharp tongue reducewith decrease in the flow rate. Because the tongue negative incidence increases withthe decrease in compressor flow rate, the sharp tongue blockage increases fasterthan the rounded tongue blockage. Figure 3.40 shows the nondimensional turbulent



1.040

1.045

1.050

0 50 100 150 200 250 300 350

Circumferential angle (deg)

Cp

Cp volute inlet hub (experimental)

Cp volute inlet hub (Algebraic model)

Cp volute inlet hub (K-E model)

Figure 3.38. Static pressure distribution along volute inlet hub wall.

0.6

0.55

0.5

0.6 0.8 1 1.2 1.4

0.45

0.4

0.35

ψ

r/b2 = 0.3

r/b2 = 0.0

φ/φ0

Figure 3.39. Performance characteristic.

viscosity (µT /µ) distributions obtained by employing the k-ε turbulence equations.It can be seen that, for a lower flow rate and design flow rate cases, the turbu-lent viscosity is uniformly distributed excepted near the tongue area. For the lowerflow rate, we can see a higher turbulence level behind the diffuser vane. Therefore,in a flow rate near surge and design, the zero-equation turbulence model can providea reasonable prediction of flow characteristics.



φ/φ0 = 0.75 φ/φ0 = 1.07 φ/φ0 = 1.25

Turbulence viscosityTurbulence viscosity

Turbulence viscosity

10065

2000

1525

1050

575

100[Pa s]

60

55

50

45

75

50

25

0[Pa s]

Figure 3.40. Turbulent viscosity (µτ /µ) contours at midplane of volute.

(a) (b) (c)

Figure 3.41. Velocity vector at θ= 180 degrees (φ/φ0 = 1.07). (a) Sharp tongue(zero-equation model) (b) Rounded tongue (zero-equation model)(c) Rounded tongue (k-ε model).

(a) (b) (c)

Figure 3.42. Velocity vector at scroll exit (θ= 360 degrees (φ/φ0 = 1.07). (a) Sharptongue (zero-equation model); (b) Rounded tongue (zero-equationmodel); (c) Rounded tongue (k-ε model).

In the investigation of the tongue shape influence of the compressor, the sharptongue scroll model was analyzed by using the zero-equation turbulence model.Figures 3.41 and 42 show secondary flow vectors at cross sections θ= 180 and 360degrees for a sharp tongue and a rounded tongue scrolls at a design flow condition. Itcan be seen that the secondary flow in the scroll sections has a single vortex structure



(a) (b)

TongueTongue

Figure 3.43. Velocity vectors near tongue intersection at midplane (φ/φ0 = 1.07).(a) Sharp tongue. (b) Rounded tongue.

in both tongue geometries. It can also be seen that the tongue shape changes thesecondary flow structures. The secondary flow impacted by the tongue occurs notonly near the tongue area but also in other sections of the scroll. The rounded tonguecreates a relatively large blockage near the tongue area.The large blockage forces thesecondary flow center away from the tongue area. At both θ= 180 and 360 degreelocations, the secondary flow is stronger for the rounded tongue compared withthe sharp tongue. The secondary flow center for sharp tongue occurs in the regioncloser to the tongue compared with that for the rounded tongue. As can be observedin these figures, the difference between the two turbulence models is very small.

The flow structures near the tongues are similar for both tongues at a design flowcondition seen in Figure 3.43. The flow leaves the tongue leading edge smoothlyat a design flow rate for both cases. No separation zone is found. It is also foundthat the sharp tongue has a smaller blockage than the rounded tongue (Figure 3.43).The flow near the leading edge of the rounded tongue is blocked by the tongue andthen pushed away from the tongue.

Figures 3.44 and 45 show computed static pressure contours for the scroll withrounded tongue near the design flow rate at cross sections θ= 180 degree and 360degree at a design condition with different turbulence models. Figure 3.46 shows theexit plane total pressure contours at the design condition. It can be seen that the k-εturbulence model predicted lower static pressure and higher total pressure. However,the differences of the mass average pressures are less than 1%, as indication inFigure 3.46. The total pressure distributions are asymmetrical for both cases, andthe symmetrical total pressure distributions are caused by secondary flows. Thelocation of the highest total pressure range is different for the two calculations.It is attributed to the usage of the different turbulence models, which resulted indifferent secondary flow predictions.

Figure 3.47 shows the velocity vectors near the tongue intersection at a midplaneand the secondary velocity vectors near the tongue trailing edge near the chokeflow rate φ/φ0 = 1.25. It is observed that the flows separate near the tongue for both



(a) (b)

350000349786349571349357349143348929348714348500348286348071347857347643347429347214347000

[Pa]

Pressure Pressure350000349786349571349357349143348929348714348500348286348071347857347643347429347214347000

[Pa]

Figure 3.44. Static pressure contour at θ= 180 degree (φ/φ0 = 1.07). (a) Zero-equation model. (b) k-ε model.

349,000348,786348,571348,357348,143347,929347,714347,500347,286347,071346,857346,643346,429346,214346,000

[Pa]

Pressure

(a) (b)

349,000348,786348,571348,357348,143347,929347,714347,500347,286347,071346,857346,643346,429346,214346,000

[Pa]

Pressure

Figure 3.45. Static pressure contour at scroll exit (θ= 360 degree (φ/φ0 = 1.07).(a) Zero-equation model. (b) k-ε model.

Total pressure

(a) (b)

Total pressure350,300

350,200350,150350,100350,050350,000359,950359,900359,850359,800359,750359,700359,650359,500

350,250350,300

350,230

350,160

350,090

350,020

349,950

349,880

349,810

349,740

349,670

349,600

Figure 3.46. Total pressure contour at discharge of the exit cone (φ/φ0 = 1.07).(a) Zero-equation model. (b) k-ε model.



(a) (b)

Figure 3.47. Velocity vectors near tongue intersection at midplane (φ/φ0 = 1.25).(a) Zero-equation model. (b) k-ε model.

Total pressure

(a) (b)[Pa]

250,000

249,444

248,889

248,333

247,778

247,222

246,667

246,111

245,556

245,000

Total pressure

[Pa]

250,000

249,444

248,889

248,333

247,778

247,222

246,667

246,111

245,556

245,000

Figure 3.48. Total pressure contours near tongue intersection at midplane(φ/φ0 = 1.25). (a) Zero-equation model. (b) k-ε model.

cases. The scroll tongue encounters a large incidence because high flow rate resultsin large flow angles from the tangential direction. A large incidence results in flowseparations from near the leading edge of the tongue. The k-ε turbulence modelpredicts a recirculation zone near the inlet of the volute, and the zero-equationturbulence model predicts a larger recirculation zone. Total pressure contours nearthe tongue intersection at midplane are shown in Figure 3.48.

Total pressure distributions near the tongue are different between two differentturbulence model predictions. The zero-equation model predicts the lowest totalpressure zone near the tongue. However, the k-ε turbulence model predicts thelowest total pressure in the region, which is located near the volute inlet tongue area.



Pressure

[Pa]

145,000144,786144,571144,357144,143143,929143,714143,500143,286143,071142,857142,643142,429142,214142,000

Pressure

(a) (b)[Pa]

145,000144,786144,571144,357144,143143,929143,714143,500143,286143,071142,857142,643142,429142,214142,000

Figure 3.49. Static pressure at choke flow at θ= 180 degrees (φ/φ0 = 1.25).(a) Zero-equation model. (b) k-ε model.

Pressure Pressure

140,000139,786139,571139,357139,143138,929138,714138,500138,286138,071137,857137,643137,429137,214137,000

140,000139,786139,571139,357139,143138,929138,714138,500138,286138,071137,857137,643137,429137,214137,000

[Pa] [Pa](a) (b)

Figure 3.50. Static pressure at choke flow at θ= 360 degree (φ/φ0 = 1.25).(a) Zero-equation model. (b) k-ε model.

The computed static pressure contours near the choke flow rate at cross sectionsθ= 180 and 360 degrees are shown in Figures 3.49 and 50. It can be seen that thek-ε model predicted higher static pressure level than the zero-equation model. Thetotal pressure distributions at the scroll exit are shown in Figure 3.51. The totalpressure level predicted using the k-ε model is higher than that using the zero-equation model. The lowest total pressure zone location appears to be different forthe two computational cases. This is because these two turbulence models result indifferent strengths of the secondary flow. The total pressure distributions predictedusing the k-ε model showed the boundary-layer effects; that is, there is a lowertotal pressure zone around the discharge pipe wall. Figure 3.52 shows velocityvectors near the tongue intersection at midplane near the surge flow rateφ/φ0 = 0.75.Flow vectors near the tongue indicate a continuous flow passing through the tongueand into the exit duct. Flow vectors also indicate that due to the large recirculationflow, the pressure side of the tongue does not show significant flow separation.The large cutback tongue with significant flow circulation inside the scroll part candelay the flow separation near scroll tongue at a small flow rate. Two types of the



Total pressure

(a) (b)

[Pa]

145,300

144,893

144,486

144,079

143,671

143,264

142,857

142,450

142,043

141,636

141,229

140,821

140,414

140,007

139,600

Total pressure

[Pa]

145,300

144,893

144,486

144,079

143,671

143,264

142,857

142,450

142,043

141,636

141,229

140,821

140,414

140,007

139,600

Figure 3.51. Total pressure discharge of exit cone at choke flow (φ/φ0 = 1.25).(a) Zero-equation model. (b) k-ε model.

(a) (b)

Figure 3.52. Velocity vectors near intersect diffuser and tongue near surge condi-tion (φ/φ0 = 0.75). (a) Zero-equation model. (b) k-ε model.

turbulence model provided very similar results. Total pressure distributions nearthe scroll tongue at the midplane area are shown in Figure 3.53. Total pressuredistributions show that the k-ε model predicted a slightly higher level of the totalpressure. The performance test also confirms this. Static pressure contours at crosssections θ= 180 and 360 degrees are shown in Figures 3.54 and 55. It is shownthat the k-ε model predicted a lower level of static pressure than the zero-equationturbulence model. Total pressure distributions as shown in Figure 3.56 indicate thatthe k-ε model predicted higher total pressure levels than the zero-equation model.However, the differences between two calculations are insignificantly small. Again,the boundary-layer effects can be seen in the k-ε turbulence model calculations.

The calculations showed that, by using different turbulence models, the perfor-mance prediction results did not make significant difference. In these calculations,the impeller and the vaned diffuser as well as scroll tongue have relatively sharp



(a) (b)

Figure 3.53. Total pressure contours near tongue intersection at midplane(φ/φ0 = 0.75). (a) Zero-equation model. (b) k-ε model.

[Pa]

[Pa]

Pressure

(a) (b)

Pressure (ps180)

405,000

404,786

404,571

404,357

404,143

403,929

403,714

403,500

403,286

403,071

402,857

402,643

402,429

402,214

402,000

405,000

404,786

404,571

404,357

404,143

403,929

403,714

403,500

403,286

403,071

402,857

402,643

402,429

402,214

402,000

Figure 3.54. Static pressure at surge condition at θ= 180 degrees (φ/φ0 = 0.75).(a) Zero-equation model. (b) k-ε model.

[Pa]

Pressure

405,000404,786404,571404,357404,143403,929403,714403,500403,286403,071402,857402,643402,429402,214402,000

[Pa]

Pressure (ps360)

3.490e+05

3.487e+05

3.484e+05

3.481e+05

3.478e+05

3.475e+05

3.472e+05

3.469e+05

3.466e+05

3.463e+05

3.460e+05

(a) (b)

Figure 3.55. Static pressure at surge flow at θ= 360 degrees (φ/φ0 = 0.75).(a) Zero-equation model. (b) k-ε model.



[Pa]

Total pressure

(a) (b)

405,800

405,814

405,829

405,843

405,857

405,871

405,886

405,900

405,914

405,929

405,943

405,957

405,971

405,986

406,000

[Pa]

Total pressure

405,800

405,814

405,829

405,843

405,857

405,871

405,886

405,900

405,914

405,929

405,943

405,957

405,971

405,986

406,000

Figure 3.56. Total pressure discharge of exit cone at surge flow (φ/φ0 = 0.75).(a) Zero-equation model. (b) k-ε model.

leading edges. The flow losses may mainly be dependent on the flow angleseparations; this type of flow separation is not sensitive to the turbulence models.

3.10 CFD Applications in Turbomachine Design

With the development of the computer hardware, the CFD has been implementedin the turbomachine design process. Enormous efforts are devoted to improving theefficiency of the gas turbine components. The designs for turbine and compressorairfoils play one of the most important roles in increasing the turbine efficiency. Inthe airfoil design, there are two types of implementations the aerodynamic designengineer often considers. One is to design and employ custom-design blade profileswith minimum losses and controlled blade boundary-layers. The second and evenmore complex part is to minimize losses resulting from secondary flows near huband casing. Recently three-dimensional blade design concepts that help to controlsecondary flows were proposed [1]. However, the complex flow is difficult even withfully three-dimensional Navier–Stokes flow solvers. And the validation of the N–Ssolvers need to take large numbers of experimental data, which is time consumingand expensive. Therefore, almost all the aerodynamic designs are based on the two-dimensional design. The inviscous analyses of the two-dimensional airfoil sectionsstill play an important role in the design process.

It is known that for a blade row in an annulus, the stream surface between twoannular walls is twisted for most cases. These twists are induced by either shedvorticity or Secondary flow arising from. Stream surface twist can arise in an irro-tational flow owing to either spanwise components of velocity or spanwise bladeforces. Many efforts have been taken to reduce the stream surface twist and thesecondary flow losses, such as sweep, lean, bow, and twist the blades or design anasymmetric end wall. However, there is little information available in the literaturefor using three-dimensional design and almost no information available to showhow to integrate the three-dimensional features into the design process. Most of the



turbomachinery design system investigations were still on the academic researchand were based on the particular machines or blades. Moreover, most of the studieswere based on the particular blade and flow situations. For example, Singh et al.[14] argued that closing the blade throat near the end walls could obtain significantefficiency improvements, and Wallis and Denton [20] also obtained an efficiencyincrease from almost the opposite type of blade twist near the end wall. For dif-ferent machines and different designs, many different techniques should be usedaccording to the flow field nature of the designs. It is very important to design ablade design procedure and optimize the design.

The increased use of CFD tools has been driven mainly by two factors. First,from performance standard point of view, efficiency has steadily increased. Sec-ond, the turbomachinery industry as a whole has been pushed toward reduced costdesigns. The cost reduction is in terms of development, modification, production,and operating costs. The cost reduction drives a turbomachine toward high loadingin order to reduce stage count, while maintaining or exceeding past performancegoals. The current design of the new stage already is outside of the standard air-foil database. Most of the airfoil needs to be designed. And the development ofthe design tools to meet this requirement becomes critical. The application of CFDmethodology to improve the turbomachinery design is becoming established withinthe turbomachinery community. However, still only limited number of publicationssuggest how to use CFD to help design and modify processes especially during theblade design. This chapter serves to present a design process that contains a noveltwo- and three-dimensional viscous turbulent code and optimization process.

Expensive manpower is invested in order to find configurations that are sta-ble and efficient in the work range in the turbine and compressor designs. One ofthe most important methods is inverse design or called stream line design wheretwo-dimensional blade profiles are to be found to insure the desired working rangestability and efficiency. During the design, the constraints arising from aerodynam-ics, aeromechanical, mechanical, heat transfer, and manufacturing considerationshave to be satisfied.

The design of turbine and compressor blade had made a great progress. Manyadvanced design methods and CFD tools have been incorporated into the designprocedure. However, most of the design procedures focus only on the flow pre-diction, and there are few papers that describe the overall design processes anddesign implements. For example, Wellborn and Delaney [5] described a com-pressor design system in Rolls-Royce, which comprises three tools: through-flow,three-dimensional isolated blade, and three-dimensional multistage predication.The turbomachinery design is an integrated process that contains a process frommeanline, through-flow, airfoil design, and analysis. This paper developed a designprocess that can be easily adapted by the industry.

The aerodynamic design procedures for turbomachinery airfoil used in this studyis shown in Figure 3.57. A design system includes meanline analysis, through-flow analysis, airfoil section design, airfoil stackup, three-dimensional blade row,and multistage flow analysis. For obtaining the highest design efficiency, theoptimizer was used to do the section optimization. The three-dimensional CFD



2D axisymmetric through-flowanalysis

Meanline analysis

3D blade row calculation

3D multistage calculation

Optimizer

Optimizer

Airfoil section design

Section viscous analysis

Airfoil stackup

Figure 3.57. Blade design and optimization procedure.

code was used for blade stackup optimization. The optimizer may be used forthree-dimensional blading although authors do not encourage the use of optimizerfor three-dimensional optimization.

Meanline analysis determines the loading of the stage and annular area. Itplays an important role in the turbomachinery design. The meanline design forthe first-stage compressor and the last-stage turbine is critical. The enthalpy risefor compressor and drop for turbine are fixed by the meanline analysis. The overallmachine character is determined by the meanline analysis. The compressor andturbine efficiency is strongly influenced by meanline design.

Through-flow analysis is one of the preliminary design modules. The axisym-metric streamline curvature calculations can be used to optimize the overallparameters of a multistage turbomachine. This module establishes the definitionsof the flow path and work distributions in radio direction. The velocity diagrams atdesign point for different blade rows and different streamlines are determined. Theoptimization code can be used to do the optimization for selecting the best designparameters, for example, stage loading and stage enthalpy change.



The airfoil section design is a very important step for the aerodynamic design.All the airfoils are designed by the section design and reasonable stackup. A three-dimensional blade method was developed in this study as shown in Figure 3.57.It can be seen that CFD code for both section analysis and three-dimensional bladerow analysis is very important. The development of the efficient and wide-rangeapplication of CFD codes is very important. In this study, an effective numericalmethod was developed for both two-dimensional and three-dimensional codes. Inthis chapter, a brief introduction for the numerical method developed for two-dimensional incompressible and compressible flows was provided.

3.10.1 Flow solver for section analysis

A time-marching algorithm was used in the present cascade flow computations[2,20,22,23,29]. The computation starts with a rough perturbation, which developsunder certain boundary conditions. In this approach, the governing equations arereplaced by a time-difference approximation with which steady or time-dependentflows of interest can be solved at each time level. The details of the governing equa-tions and the method of solving the equations are discussed in previous sections.

The discussion has been extended for many years for the optimum type of themesh that should be used for turbine or compressor blade flow calculation [15,16].The more orthogonal the grid, the smaller will be the numerical errors due tothe discretization. However, no one type of grid is ideal for blade-to-blade flowcalculation. In this study, the H-type mesh is used. Another problem for the blade-to-blade mesh is the trailing edge mesh and the trailing edge Kutta condition [7].The authors have noticed that the number of mesh points near the trailing edgepoints strongly influence the loss calculation. Here, a realistic method is proposed;that is, when the mesh on the trailing edge circle is more than 10 points, an explicitviscous Kutta condition [17] is applied on the training edge circle which allowsflow leaving the blade smoothly.

The mesh refinement study was conducted prior to the calculations [2,3]. It isshown that the mesh size of 110 × 45 with 80 points on the blade surface in the blade-to-blade section is sufficient. The H-mesh [2,3] was used in all the computations.The computational mesh is shown in Figure 3.58. A typical convergence historyof the calculation is shown in Figure 3.59. It is demonstrated that the present codehas a good convergence. In this study, this code was used in the two-dimensionalairfoil section analysis and optimization.

3.10.2 Optimization

Turbine and compressor designs faced the challenge of high efficiency and reliableblade design. Design techniques are typically based on the engineering experienceand may involve much trial and error before an accepted design is found. The abilityof the three-dimensional codes and the ability to perform a three-dimensional flowcalculation are difficult to improve the design. The three-dimensional approachis a way to help designers do some parameter study and compare the numerical



Figure 3.58. Computational mesh.

Number of steps

Err

or x

100

0

0 500 1000 1500 2000

100

10−1

10−2

10−3

Figure 3.59. Typical convergence history.

study with testing in order to improve the design. The three-dimensional method isused for parameter studies, like lean, bow, and sweep effects. The two-dimensionalanalysis developed in the study is used to optimize blade section airfoil.

The blade design often starts with two-dimensional airfoil section. Some ofthe design parameters are obtained by through-flow optimization and vibration andheat transfer analysis, for example, the solidity and number of blades. And most ofthe parameters are optimized through the sectional design. For example, leading-and trailing-edge radii, stagger angle, and maximal thickness position. The finalthree-dimensional analyses are used to make lean, bowed, and sweep modificationof the blades.

The optimization process used in this study was based on the constrained opti-mization method [26]. If the objective function to be minimized is F(X) andconstraint functions is gj(X), the optimization problem can be formulated by the



objective and constraint functions as:

∇F(X) +∑λj∇gj(X) = 0 (78)

where λj are the Lagrange multipliers and X is the vector of the objective. Thefinite differencing method can be used to obtain the gradients of objective and con-straint functions with respect to design variables and constraints. Many commercialoptimization packages are available as an optimizer to design problems [26].

The objective function of this study is adiabatic efficiency. The optimizationobjective is to obtain the highest efficiency under given constraints. Foe moreconvenience, the total pressure loss is used as an objective variable to judge thedesign. The definition of the total pressure loss in this study is:

ξ = (p∗in − p∗

out)

H(79)

where H is the outlet dynamic head of the exit plane.

Two-dimensional section optimizationIn this study, a compressor stator blade originally stacked up by using NACA 65was redesigned and optimized. In this design, the thickness of the airfoil, chord,stagger angle, and inlet and outlet flow parameters were fixed. The designed bladewas used to replace the old blade. The section design optimization was mainlyachieved through adjusting the section maximum thickness location. It was foundthat the maximum thickness locations influence the losses and performance. In thisstudy, five sections were selected as design section. All section designs used thesimilar method and procedure. As an example, the results presented here are thedesign information for 50% span section.

Figures 3.60 through 62 show the Mach number distributions of the designblades with different maximum thickness locations. Figure 3.63 shows the relation-ship between total pressure losses and location of the maximum thickness. It wasshown that the Mach number distributions changed with the maximum thickness.The change in the Mach number distributions will influence the boundary-layerdevelopment and influence the losses of the section. It was found that there is an opti-mum location of the maximum thickness. Studies have also shown that this locationchanges with flow conditions and stagger of the airfoil. After optimization study, theairfoil section with highest efficiency was selected. The airfoil was stacked up usinga smooth leading edge curve method. Before the airfoil was selected as the finalblade, the three-dimensional analysis was performed to investigate the benefits ofthe three-dimensional blade features. Three-dimensional analysis showed that aftersection optimization the airfoil performance was improved, as shown in Figure 3.64.However, the end wall region did not show significant improvement. Some three-dimensional features were used to reduce the losses and increase the efficiency.

Three-dimensional bladingThree-dimensional CFD analysis codes were used to guide the three-dimensionalfinal modification of the two-dimensional section stackup. Some parameters were



0.65

0.75

0.75

0.30.65

0.15

0.05

0.350.65

0.85

0.450.35

0.55

0.05

0.55

0.25

0.55

0.550.

45

Figure 3.60. Mach number distributions for maximum thickness at 15% chord.

0.65

0.55

0.35

0.55

0.65

0.45

0.75

0.350.25

0.65

0.45

0.15

0.25

0.05

0.55

0.25

0.050.35

0.65

0.550.15

0.25

0.45

Figure 3.61. Mach number distributions at maximum thickness at 25% chord.



Figure 3.62. Mach number distributions at maximum thickness at 45% chord.

0.35

0.4

0.45

0.5

0.55

15 25 35 45Maximum thickness position (%)

Tota

l pre

ssur

e lo

ss (

%)

Figure 3.63. Total pressure losses Vs maximum thickness location.

selected to do the study based on design experience. In this redesign, the three-dimensional feature-bowed blade was used to reduce the secondary losses. Inthe three-dimensional blading, the optimizer will not be used in this study,because optimization with optimizer like the three-dimensional code is very timeconsuming. Although the mesh sizes, turbulence models and mixing between theblade rows strongly influence the optimization results, the process propose hereprovide a way to drive a better design. In the current study, three-dimensionaloptimization was obtained through parameter study.



0

10

20

30

40

50

60

70

80

90

100

0 3 6 9 12 15

Total pressure loss (%)

Rad

ial h

eigh

t (%

) After optimization

After redesign sections

NACA 65 sections

Figure 3.64. Total pressure losses along with blade span.

(a) (b) (c)

Figure 3.65. Static pressure distribution on the suction surface. (a) Original NACAsection blade. (b) Before 3D optimization. (c) After 3D optimization.

In this study, only bow feature is applied. The bow location and degree of thebow were selected as study parameters. It was found that the 15 degree bow locatedat 30% of the span can eliminate separation and get better performance. The three-dimensional analysis showed that after two-dimensional section optimized design,the flow in most of the spanwise locations remained attached and well behavedalthough it had a small region of end wall separation on the section side, as shownin Figures 3.65 through 68. After the bow was introduced the flow around theairfoil on both tip and root end wall was attached. The total pressure losses werereduced, as shown in Figure 3.64. This study showed that the total section efficiencyincreased by about 3% as compared with the original blade. Both two-dimensional



(a) (b) (c)

Figure 3.66. Axial velocity contour near blade suction surface. (a) Original NACAsection blade. (b) Before 3D optimization. (c) After 3D optimization.

(a) (b)

Figure 3.67. Velocity vector near the root section near the trailing edge suctionsurface. (a) Before 3D optimization. (b) After 3D optimization.

section design and three-dimensional optimization increased the efficiency withsimilar percentage. It has been shown that both two-dimensional optimizationand three-dimensional study are important. However, three-dimensional bladingdepends more on the experience of the designer.

It is important to point out that, based on the design experience, the three-dimensional analysis results normally are not accepted as final shape. The selectionof the final configuration was accepted through small adjustments based on expe-rience to obtain high performance and stability. After three-dimensional analysis,some small adjustments were made based on the possible manufacturing uncertain-ties and applications. For example, the compressor stator design in this study wasrecambered to increase exit angle by about 1.5 degrees to increase the compressorsurge margin.



(a) (b)

Figure 3.68. Velocity vector near the tip section near the trailing edge suctionsurface. (a) Before 3D optimization. (b) After 3D optimization.

The flow field around turbine and compressor blades exhibits very complexflow features. The flow field involves various types of loss phenomena, and hencehigh-level flow physics is required to produce reliable flow predications. The bladedesign process is a very time consuming process and the optimization process isvery complicated. The method developed in this study for the airfoil section andblade design and optimization is one successful process and very easy for adaptingin the aerodynamic design. In this study, a two-dimensional code was used forsection design instead of three-dimensional code, which provides an economic wayfor design and optimization. The three-dimensional blades were optimized afterstacking up the two-dimensional sections. The lean, bowed, and swept features ofthe three-dimensional blade were created during three-dimensional optimization.It was shown that both two- and three-dimensional design and analysis were thebackbones of the blade aerodynamic designs. The results show that both the airfoilshape optimization and the three-dimensional optimization can be used to improvethe performance of compressor and turbine blade.

References

[1] Xu, C., and Amano, R. S. On the Development of Turbine Blade Aerodynamic DesignSystem, ASME and IGTI Turbo Expo, 2001-GT-0443. New Orleans, LA, USA, June4–7, 2001.

[2] Xu, C., and Amano, R. S. A Hybrid Numerical Procedure for Cascade Flow Analysis,Numer. Heat Transfer, Part B, 37(2), pp. 141–164, 2000.

[3] Xu, C., and Amano, R. S. An Implicit Scheme for Cascade Flow and Heat TransferAnalysis, ASME J. Turbomach., 122, pp. 294–300, 2000.

[4] Japikse, D. Centrifugal Compressor Design and Performance, Concepts ETI, Inc.,Wilder VT, USA, 1996.

[5] Wellborn, S. R., and Delaney, R. A. Redesign of a 12-Stage Axial-Flow CompressorUsing Multistage CFD, 2001-GT-345, ASME and IGTITurbo Expo, New Orleans, LA,USA, June 4–7, 2001.



[6] Lakshminarayana, B. An Assessment of Computational Fluid Dynamics Techniquesin the Analysis and Design of Turbomachinery—the 1990 Freeman Scholar Lecture,ASME J. Fluids Eng., 113, pp. 315–352, 1991.

[7] Xu, C. Kutta Condition for Sharp Edge Flows, Mech. Res. Commun., 25(4), pp.425–420, 1998.

[8] Xu, C. Surface Vorticity Modeling of Flow Around Airfoils, Comput. Mech., 22(6),pp. 1998.

[9] Xu, C., and Yeung, W. W. H. Discrete Vortex Method for Airfoil with UnsteadySeparated Flow, J. Aircraft, 33(6), pp. 1208–1210, 1996.

[10] Xu, C., Yeung, W. W. H., and Gou, R.W. Inviscid and Viscous Simulations of SpoilerPerformance, Aeronaut. J., 102(1017), pp. 399–405, 1998.

[11] Xu, C., and Yeung, W. W. H. Unsteady Aerodynamic Characteristics of Airfoil withMoving Spoilers, J. Aircraft, 36(3), pp. 530–538, 1999.

[12] Xu, C., and Yeung, W. W. H. Numerical Study of Unsteady Flow Around Airfoil withSpoiler, 65(1), 164–171, 1998.

[13] Michos, A., Bergeles, G., and Athanassiadis, N. Aerodynamic Characteristics ofNACA0012 Airfoil in Relation to Wind Generators, Wind Eng., 7, pp. 1–8, 1985.

[14] Singh, G., Walker, P. J., and Haller, B. R. Development of 3D Stage ViscousTime-Marching Method for Optimisation of Short Stage Heights, Proceedings of theEuropean Conference on Turbomachinery, Erlangen, 1995.

[15] Turkel, E., and Vatsa, V. N. Effect of Artificial Viscosity on Three-Dimensional FlowSolutions, AIAA J., 32(1), pp. 39–45, 1994.

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II. Finite Element Method


4 The finite element method: discretizationand application to heat convection problems

Alessandro Mauro1,2, Perumal Nithiarasu1, Nicola Massarotti2,and Fausto Arpino3

1Civil and Computational Engineering Centre, School of Engineering,Swansea University, Swansea, U.K.2Dipartimento per le Tecnologie, Università degli Studi di Napoli“Parthenope”, Napoli, Italy.3Dipartimento di Meccanica, Structture, Ambiente e Territorio,Università di Cassino, Cassino, Italy.

Abstract

In this chapter, we give an overview of the finite element method and its applicationsto heat and fluid flow problems. An introduction to weighted residual approxima-tion and finite element method for heat and fluid flow equations are presented.The characteristic–based split (CBS) algorithm is also presented for solving theincompressible thermal flow equations. The algorithm is based on the temporal dis-cretization along the characteristics, and the high-order stabilization terms appearnaturally from this kind of discretization. The artificial compressibility (AC) and thesemi-implicit (SI) versions of the CBS scheme are presented and some examplesare also given to demonstrate the main features of both the schemes.

Keywords: Characteristic-based split, Finite elements, Porous media

4.1 Governing Equations

4.1.1 Non-dimensional form of fluid flow equations

In many heat transfer applications, it is often easy to generate data by non-dimensionalizing the equations, using appropriate non-dimensional scales. Toobtain a set of non-dimensional equations, let us consider five different cases ofconvection heat transfer. We start with the mixed convection problem followedby the natural and forced convection problems. The turbulent flow and convectiveflow through porous media are also presented. For each case, we discuss one set ofnon-dimensional scales.



Mixed convectionMixed convection involves features from both forced and natural flow conditionsthat are two limiting cases of mixed convection flow. The buoyancy effects becomecomparable to the forced flow effects at small and moderate Reynolds numbers.Since the flow is partly forced, a reference velocity value is normally known (exam-ple: velocity at the inlet of a channel). However, in mixed convection problems, thebuoyancy term needs to be added to the appropriate component of the momentumequation.

In incompressible mixed convection flow problems, the following non-dimensional scales are normally employed:

x∗i = xi

L, u∗

i = ui

uref, t∗ = t uref

L2 ,(1)

p∗ = p

ρu2ref

, T ∗ = T − Tref

Tw − Tref

where * indicates a non-dimensional quantity, L is a characteristic dimension, thesubscript ref indicates a constant reference value and Tw is a constant referencetemperature, for example, wall temperature. The density ρ and viscosity µ of thefluid are assumed to be constant everywhere and equal to the inlet value.

Using the above scales, the non-dimensional form of the incompressible mixedconvection equations, in indicial notation, are written as:

Continuity equation

∂u∗i

∂x∗i

= 0 (2)

Momentum equation

∂u∗i

∂t∗+ u∗

j∂u∗

i

∂x∗j

= −∂p∗

∂x∗i

+ 1

Re

∂

∂xj

(∂u∗

i

∂xj

)+ Gr

Re2 T∗γi (3)

where γi is a unit vector in the gravity direction.

Energy equation

∂T∗

∂t∗+ u∗

i∂T ∗

∂x∗i

= 1Re Pr

∂

∂xi

(∂T ∗

∂xi

)(4)

where Re is the Reynolds number defined as:

Re = uref L

ν(5)

Pr is the Prandtl number given as:

Pr = ν

α(6)


Applications of finite element method to heat convection problems 131

Fluid circulation

Hot, vertical plate

g

x2

x1

Figure 4.1. Natural convective flow near a hot vertical plate.

and Gr is the Grashof number given as:

Gr = gβTL3

ν2 (7)

where ν=µ/ρ is the kinematic viscosity. Appropriate changes will be necessary ifan appreciable variation in these quantities occurs in a flow field.

Note that sometimes, a non-dimensional parameter referred to as the Richardsonnumber (Gr/Re2) is also used in the literature.

Natural convectionNatural convection is a limiting case of the more general case of mixed convec-tion. Natural convection is generated only by the density difference induced by thetemperature differences within a fluid system (Figure 4.1). Because of the smalldensity variations present in these types of flows, a general incompressible flowapproximation is adopted. In most buoyancy-driven convection problems, flow isgenerated either by a temperature variation or by a concentration variation in thefluid system, which leads to local density differences.

In practice, the following non-dimensional scales are adopted for naturalconvection in the absence of a real reference velocity value:

x∗i = xi

L, u∗

i = uiL

α, t∗ = t α

L2 ,(8)

p∗ = pL2

ρα2, T ∗ = T − Tref

Tw − Tref



Upon introducing the above non-dimensional scales into the governing equa-tions, and replacing Re with 1/Pr in the non-dimensional mixed convectionequations of the previous subsection, we obtain the non-dimensional equationsfor natural convection flows.

Often, another non-dimensional number called the Rayleigh number is used inthe calculations. This is given as:

Ra = Gr Pr = gβTL3

να(9)

Forced convectionForced convection is another limiting case of mixed convection. In forced convec-tion flow problems, the non-dimensional scales are the same as the mixed convectionproblem. The non-dimensional form of the forced convection governing equationsis obtained from that of mixed convection by neglecting the buoyancy term on theright hand side of the momentum equation.

Another non-dimensional number, which is often employed in forced convec-tion heat transfer calculations is the Peclet number and is given as Pe = RePr =uref L/α.

Some examplesThe following results are obtained solving the above equations for forced, naturaland mixed convective flows by using an artificial compressibility (AC) version anda semi-implicit (SI) version of the characteristic-based split (CBS) algorithm andfinite element discretization technique that will be explained later.

The first problem considered is the backward-facing step, that is a test case com-monly used to validate numerical solutions of the incompressible Navier–Stokesequations. In fact, several data are available in literature for this problem (Denhamand Patrik [1], Aung [2], Ichinose et al. [3], de Sampaio et al. [4], Massarotti et al.[5]). The computational domain and the boundary conditions used in this workare presented in Figure 4.2a. In particular, the velocity profile at the inlet is mea-sured experimental profile (Denham and Patrik [1]). The measurement section inthe experiments, as in the present calculations, was placed 1.3 times the step heightupstream of the step. All the horizontal walls and the step have been imposed withno-slip velocity boundary conditions.

For non-isothermal cases, in addition to the above boundary conditions (Aung[2]), a non-dimensional temperature equal to one is imposed along the bottom wall,while the temperature of the inlet flow is assumed to be equal to zero. All other wallsare assumed to be adiabatic. Figure 4.2 shows some details of the computationalgrids used for this problem.

Figure 4.3 shows the horizontal velocity distribution along different verticalsections of the channel, obtained by using the AC and the SI versions of the CBSalgorithm.

The temperature profiles at two different vertical cross sections downstream ofthe step are presented in Figure 4.4 for the case of a fully developed inlet profile with



p = 0

q = 0

T = 1v = 0,u = 0,

v = 0,

v = 0, q = 0,

u = 0,

u = 0,

10h8h6h4h2h0.8h0−1.3h

y

x

h2h

4h

u = u(y)v = 0T = 0

(a)

(b)

(c)

Figure 4.2. Backward-facing step: (a) computational domain and boundary condi-tions; (b) unstructured mesh (955 nodes and 1,704 elements); (c) detailof the structured mesh near the step (4,183 nodes, 8,092 elements).

v

0 2 4 6 8

SI-CBS

y

−1.3

AC-CBS Denham and Patrik [26]

Figure 4.3. Backward-facing step. Comparison of the horizontal velocity profiles,Re = 229.

T

y/δt

0 0.25 0.5 0.75 1T

0 0.25 0.5 0.75 10

0.5

1

1.5

2

2.5

3AC-CBSSI-CBSAung [2]

AC-CBSSI-CBSAung [2]

x/Xr = 0.482 x/Xr = 0.946

y/δt

0

0.5

1

1.5

2

2.5

3

Figure 4.4. Backward-facing step. Temperature profiles at two different sectionsdownstream of the step (Re = 233).



u1 = u2 = 0

u1 = u2 = 0

T = 1 T = 0

Insulated

Insulated

u 1 =

u2

= 0

u 1 =

u2

= 0

Figure 4.5. Natural convection in a square cavity. Computational domain andboundary conditions.

Re = 233. The profiles are practically the same for both the schemes and they showa very good agreement with the experimental data available from the literature forthis problem (Aung [2]). It should be observed that the position along the verticalsection is related to the thickness of the thermal boundary layer in the channel.

Another benchmark problem that is commonly employed for code validation is awell known natural convection problem in which the fluid-filled square cavity withthe vertical walls subjected to different temperatures is used. The two horizontalwalls are assumed to be adiabatic as shown in Figure 4.5. No-slip velocity conditionsare applied on all walls. The flow is driven by the buoyancy forces acting in thevertical direction, caused by the temperature differences induced in the fluid. Bothstructured and unstructured meshes are used (Figure 4.6).

Figures 4.7–4.9 show the temperature contours and streamlines obtained withthe AC-CBS algorithm at different Rayleigh numbers.

The transient flow over a circular cylinder is a popular test case for validatingtemporal discretization of numerical schemes. The inlet flow is uniform and thecylinder is placed at a centreline between two slip walls. The distance from theinlet to the centre of the cylinder is 4D, where D is the diameter of the cylinder.Total length of the domain is 16D. No slip condition is applied on the cylindersurface. The problem has been solved both on two and three dimensions. Figure4.10 shows the mesh used for two-dimension case. As seen, the mesh close tothe cylinder is very fine in order to capture the boundary layer and separation.Figure 4.11 shows the contours of the horizontal and vertical velocity componentsat a real non-dimensional time of 98.02. The time history of the vertical velocitycomponent at an exit central point is plotted with respect to time and is shown inFigure 4.12.

In order to show that the method works also on three dimensions, a coarsemesh with 17,382 nodes and 69,948 tetrahedral elements was used to recomputethe solution. This mesh was generated using the PSUE software available within the



(a) (b)

(d)(c)

Figure 4.6. Natural convection in a square cavity. (a) Mesh 1: 1,251 nodes and 2,888elements; (b) Mesh 2: 2,601 nodes and 5,000 elements; (c) Mesh 3:1,338 nodes and 2,474 elements; (d) Mesh 4: 5,515 nodes and 10,596elements.

Figure 4.7. Natural convection in a square cavity. Streamlines and isotherms atRa = 105.

School of Engineering, Swansea University (Morgan et al. [6]). The finite elementmesh and the contours of velocity components are shown in Figures 4.13 and 4.14,respectively, at a real non-dimensional time of 83.33. As seen, the solution obtainedis very similar to the one shown in Figure 4.11 for a two-dimensional case.





Figure 4.10. Flow past a circular cylinder. Finite element mesh, 7,129 nodes and13,990 elements.



(a)

(b)

(c)

Figure 4.11. Flow past a circular cylinder, Re = 100. (a) Horizontal velocitycontours; (b) vertical velocity contours; (c) temperature contours.

Adimensional time

Ver

tical

vel

ocity

0.6

0.4

0.2

0 15 30 45 60

0

−0.2

−0.4

−0.6

AC-CBSde Sampaio et al. [4]

Figure 4.12. Flow past a circular cylinder. The vertical velocity distribution at anexit point of the domain with respect to real time. Comparison withde Sampaio et al. [4].



Figure 4.13. Three-dimensional flow past a circular cylinder. Finite element mesh:17,382 nodes, 69,948 elements.

Figure 4.14. Three-dimensional flow past a circular cylinder, Re = 100; u1 velocitycontours (left); u3 velocity contours (right).

4.1.2 Non-dimensional form of turbulent flow equations

For turbulent flow computations, Reynolds averaged Navier–Stokes equations ofmotion are written in conservation form as follows:

Mean continuity equation

1

β2

∂p

∂t+ ∂(ρui)

∂xi= 0 (10)



Mean momentum equation

∂ui

∂t+ ∂

∂xj(ujui) = − ∂p

∂xi+ ∂τij

∂xj+ ∂τR

ij

∂xj(11)

where β is an AC parameter, which will be well explained in Section 2.6, ui are themean velocity components, p is the pressure, ρ is the density and τij is the laminarshear stress tensor given as:

τij = 1

Re

(∂ui

∂xj+ ∂uj

∂xi− 2

3

∂uk

∂xkδij

)(12)

The Reynolds stress tensor, τRij , is introduced by Boussinesq assumption as:

τRij = νT

Re

(∂ui

∂xj+ ∂uj

∂xi− 2

3

∂uk

∂xkδij

)− 2

3kδij (13)

In the above equations, ν is the kinematic viscosity of the fluid, νT is the turbulenteddy viscosity and δij is the Kroneker delta. The following non-dimensional scalesare used to derive the above equations:

x∗i = xi

L, u∗

i = ui

uref, t∗ = t uref

L2 , p∗ = p

ρu2ref

,

k∗ = k

u2ref

, ε∗ = εL

u3ref

, v∗T = vT

vref, v∗ = v

uref(14)

where L is a characteristic dimension and the subscript ref indicates a referencevalue. The turbulent flow solution is obtained by solving equations (10) and (11)with appropriate boundary conditions and a turbulence model, as the Spalart–Allmaras model (Spalart and Allmaras [7], Nithiarasu and Liu [8], Nithiarasu et al.[9]). The Spalart–Allmaras (SA) model was first introduced for aerospace appli-cations and currently being adopted for incompressible flow calculations. The SAmodel is another one-equation model, which employs a single scalar equation andseveral constants to model turbulence. The scalar equation is:

∂v

∂t+ ∂(uj v)

∂xj= cb1Sv+ 1

Reσ

[∂

∂xi

(1 + v) ∂v

∂xi

+ cb2

(∂v

∂xi

)2]

− cw1fwRe

[v

y

]2

(15)where

S = S + 1Re

(v/k2y2)fv2 (16)

and

fv2 = 1 − X /(1 + Xfv1) (17)



In equation (16) S is the magnitude of vorticity. The eddy viscosity iscalculated as:

vT = vfv1 (18)

where

fv1 = X 3/(X 3 + c3v1) (19)

and

X = v/v (20)

The parameter fw is given as:

fw = g

[1 + c6

w3

g6 + c6w3

]1/6

(21)

where

g = r + cw2(r6 − r) (22)

and

r = 1

Re

v

Sk2y2(23)

The constants are cb1 = 0.1355, σ= 2/3, cb2 = 0.622, k = 0.41, cw1 = cb1/k2 +(1 + cb2)/σ, cw2 = 0.3, cw3 = 2 and cv1 = 7.1. From the above equations it is clearthat the turbulent kinetic energy is not calculated here.Thus, the last term of equation(13) is dropped when SA model is employed.

Some examplesThe following results are obtained by solving the above equations using anAC CBSalgorithm and finite element discretization technique that will be described later inthis chapter.

A standard test case commonly employed for turbulent incompressible flowmodels at moderate Reynolds number is the recirculating flow past a backward-facing step. The definition of the problem is shown in Figure 4.15. The characteristicdimension of the problem is the step height. All other dimensions are defined withrespect to the characteristic dimension. The step is located at a distance of fourtimes the step height from the step. The inlet channel height is two times the stepheight. The total length of the channel is 40 times the step height. The inlet velocityprofile is obtained from the experimental data reported by Denham et al. [10]. Noslip conditions apply on the solid walls. A fixed value of 0.05 for the turbulent scalarvariable of the SA model at inlet was prescribed. The scalar variable was assumedto be zero on the walls. Both structured and unstructured meshes were employedin the calculation (Figures 4.16 and 4.17).



Parabolic u1 and u2 = 0

2L

4L 36L

p = 0

L

u1 = u2 = 0

Figure 4.15. Turbulent flow past a two-dimensional backward-facing step. Problemdefinition.

Figure 4.16. Detail of the structured mesh (8,092 elements and 4,183 nodes) nearthe step.

Figure 4.17. Detail of the unstructured mesh (8,662 elements and 4,656 nodes)near the step.

Figure 4.18 shows the comparison of velocity profiles against the experimentaldata of Denham et al. [10]. The SA model predicts accurately the recirculationregion.

The other turbulence problem considered is a complex three-dimensional modelof flow through an upper human airway (Nithiarasu and Liu [8], Nithiarasu et al.[9]). The geometry used is a reconstruction of an upper human airway employedin the spray dynamics studies (Gemci et al. [11]). It is apparent from the availablestudies on particle movement in the upper human airways that this problem isimportant (Li et al. [9, 10], Martonen et al. [11]). To understand the mechanismbehind many upper human airway-related problems including “sleep apnoea” andvocal cord-related problems. Most of the reported studies on upper airway fluiddynamics use either structured or semi-structured meshes in the calculations. Here,the results obtained with an unstructured mesh are presented. The surface meshof the grid used in the present study is shown in Figure 4.19. This mesh containsjust under a million tetrahedral elements. The mesh is generated using the PSUEcode (Morgan et al. [6]). The Reynolds number of the flow is defined based on the



Exp.SA model

Re = 3,0253

2.5

1.5

Ver

tical

dis

tanc

e

Horizontal velocity

0.5

1

00 2 4 6 8 10 12 14

2

Figure 4.18. Incompressible turbulent flow past a backward-facing step. Velocityprofiles at various downstream sections at Re = 3,025.

Figure 4.19. Incompressible turbulent flow through a model upper human airway.Surface mesh.

diameter of the narrow portion close to the epiglottis. The total non-dimensionallength of the domain in the horizontal direction is 29.68, and in the vertical directionit is 23.05. The diameter at the inlet of the geometry (at the top) is 4.91.

A uniform velocity is assumed at the inlet of the geometry in the negativedirection perpendicular to the inlet surface in the downward direction. No slipconditions are assumed on the solid walls. The turbulent scalar variable value at theinlet is fixed at 0.05 and assumed to be equal to zero on the walls.

Figure 4.20 shows the contours of velocity components and pressure within asection along the axis in the middle of the geometry. It is apparent that the majorityof the activities are taking place near the narrow portion close to the epiglottis ofthe upper human airway. The flow is accelerated as it passes through the narrowportion of the airway. As seen, the pressure contours are clustered close to thenarrow portion representing a very high gradient region. Figure 4.21 shows thevelocity vectors within the section. The velocity vectors close to the narrow portionare also shown in this figure. As seen, the recirculation region is clearly predictedby the AC-CBS method.



(a) (b)

(c)

Figure 4.20. Incompressible turbulent flow through a model upper human airway.(a) u1 contours, (b) u3 contours, (c) pressure contours.

(a) (b)

Figure 4.21. Incompressible turbulent flow through a model upper human airway.(a) Velocity vectors and (b) velocity vectors near the narrow portion.

4.1.3 Porous media flow: the generalized model equations

The general form of the equations for a porous medium can be derived by averagingthe Navier–Stokes equations over a representative elementary volume (REV; Figure4.22), using the well-known volume averaging procedure (Whitaker [15], Vafai andTien [16], Hsu and Cheng [17]).



Fluid

REV

Figure 4.22. Representative elementary volume.

The non-dimensional form of the generalized model for the description of flowthrough a fluid-saturated porous medium can be written as:

Continuity equation

∂u∗i

∂x∗i

= 0 (24)

Momentum equation

1ε

∂u∗i

∂t∗+ 1

ε2u∗

j∂u∗

i

∂x∗j

= −∂p∗f

∂x∗i

+ J

εRe

∂

∂x∗j

(∂u∗

i

∂x∗j

)(25)

− 1

Re Dau∗

i − F√Da

|u∗|u∗i + Ra

Pr Re2 T∗γi

Energy equation

σ∂T ∗

∂t∗+ u∗

i∂T ∗

∂x∗i

= λ∗

Pr Re

∂

∂x∗i

(∂T ∗

∂x∗i

)(26)

where Da is the Darcy number and F is the Forchheimer coefficient. The buoyancyeffects are incorporated by invoking the Boussinesq approximation:

g(ρf − ρref ) = ρref gβ(Tref − T ) (27)

where the subscript f refers to the fluid that saturates the porous medium. The scalesand the parameters used to derive the above non-dimensional equations for mixedconvection through a saturated porous medium are the same as those shown at thebeginning of the chapter. The new parameters introduced here are:

J = µeff

µf, σ = ε(ρcp)f + (1 − ε)(ρcp)s

(ρcp)f, λ∗ = λeff

λf, Da = κ

L2 (28)

where J is the ratio between the effective and the fluid viscosity, σ is the heat capac-ity ratio, λ* is the ratio between the effective and the fluid thermal conductivity, Da is



v = parab.

v = 0u = 0q = 1

v = 0u = 0q = 1

x

y

u = 0T = 0

p = 0

2

(a) (b)

Figure 4.23. Mixed convection in porous vertical channel. (a) Computationaldomain and boundary conditions; (b) detail of the structured com-putational grid near the entrance (8,601 nodes and 16,800 elements).

the Darcy number, κ and ε are the permeability and porosity of the medium, respec-tively, the subscripts p and eff refer to the porous medium and to the effective values,respectively. As mentioned, equations (24)–(26) are derived for mixed convectionproblems, therefore this set of PDEs describes both natural and forced convection.When forced convection dominates the problem (Ra/PrRe2< 1), the buoyancy termon the right hand side of the momentum conservation can be neglected.

The generalized model equations introduced above reduce to the Navier–Stokesequations when the solid matrix in the porous medium disappears, that is whenε→ 1 and Da → ∞, while, as the porosity ε→ 0 and Da → 0, the equations rep-resent a solid. Therefore, the procedure can be used to describe interface problemsin which a saturated porous medium interacts with a single-phase fluid. In thesecases, it is possible to use a single domain approach, by changing the property ofthe medium accordingly.

Some examplesThe following results are obtained by solving the above equations for mixed con-vection flows by using an AC version of the CBS algorithm and finite elementdiscretization technique that will be explained later.

The example showed here is the fully developed mixed convection in a regionfilled with a fluid-saturated porous medium, confined between two vertical walls.The computational domain and the boundary conditions are shown in Figure 4.23a.



x

u/u

mea

n

−1 0 10

1

2 Chen et al. [18]PresentAC-CBS

Ra = 0Ra = 104

Ra = 2x104

Ra = 5x104

Figure 4.24. Mixed convection in porous vertical channel. Non-dimensionalvertical velocity at different Ra for aiding flow, at Da = 10−4.

The flow enters the domain from the bottom at a non-dimensional temperatureof T = 0 with a fully developed parabolic velocity profile. The uniform heat fluxcondition is symmetrically imposed on both walls. Figure 4.23b shows the detailsof the computational grid near the entrance. The mesh employed has 8,601 nodesand 16,800 elements and it is refined near the walls and at the inlet region.

Figure 4.24 shows the dimensionless vertical velocity profile at a Darcy numberequal to 10−4 and four different Rayleigh numbers.The present results are comparedwith the numerical results of Chen et al. [18]. For pure forced convection (Ra = 0),the velocity profile is flat in the central part of the domain and sharply goes tozero at the walls of the channel. In the case of aiding flow, if the Rayleigh numberincreases, the buoyancy forces result in an increase in the fluid velocity near thewalls. As the flow rate entering the channel is the same for any Rayleigh numbersconsidered, the increase in fluid velocity in the near-wall regions will result in adecreased velocity at the centre region of the channel. When Rayleigh number isincreased to 5 × 104, the maximum velocity in the near-wall region is significantlyhigher than the fluid velocity at the centre of the channel.

4.2 The Finite Element Method

We provide here an introduction to weighted residual approximation and finiteelement method for self-adjoint equations and for convection–diffusion equations.

4.2.1 Strong and weak forms

The Laplace equation is a very convenient example for the start of numerical approx-imations. We shall generalize slightly and discuss in some detail the quasi-harmonic



(Poisson) equation:

− ∂

∂xi

(k∂φ

∂xi

)+ Q = 0 (29)

where k and Q are specified functions. These equations together with appropriateboundary conditions define the problem uniquely. The boundary conditions can beof Dirichlet type:

φ = φ, on φ (30)

or that of Neumann type:

qn = −k∂φ

∂n= qn, on q (31)

where a bar denotes a specified quantity. Equations (29)–(31) are known as thestrong form of the problem.

We note that direct use of equation (29) requires computation of second deriva-tives to solve a problem using approximate techniques. This requirement may beweakened by considering an integral expression for equation (29) written as:∫

v

[− ∂

∂xi

(k∂φ

∂xi

)+ Q

]d = 0 (32)

in which v is an arbitrary function.If we assume equation (29) is not zero at some point xi in then we can also let

v be a positive parameter times the same value resulting in a positive result for theintegral equation (32). Since this violates the equality we conclude that equation(29) must be zero for every xi in , hence proving its equality with equation (32).

We may integrate by parts the second derivative terms in equation (32) to obtain:∫

∂v

∂xi

(k∂φ

∂xi

)d+

∫

vQd−∫

vni

(k∂φ

∂xi

)d = 0 (33)

We now split the boundary into two parts, φ and q, with =φ+q, anduse equation (31) in equation (33) to give:∫

∂v

∂xi

(k∂φ

∂xi

)d+

∫

vQd−∫q

vqnd = 0 (34)

which is valid only if v vanishes on φ. Hence we must impose equation (30) forequivalence.

Equation (34) is known as the weak form of the problem since only first deriva-tives are necessary in constructing a solution. Such forms are the basis for obtainingthe finite element solutions.



4.2.2 Weighted residual approximation

In a weighted residual scheme, an approximation to the independent variable φ iswritten as a sum of known trial functions (basis functions) Na(xi) and unknownparameters φa. Thus we can always write:

φ ≈ φ = N1(xi)φ1 + N2(xi)φ2 + . . .(35)

=n∑

a=1

Na(xi)φa = N(xi)ϕ

where

N = [N1, N2, . . .Nn] (36)

and

φ = [φ1, φ2, . . . , φn]T

(37)

In a similar way we can express the arbitrary variable v as:

v ≈ v = W1(xi)v1 + W2(xi)v2 + . . .

=n∑

a=1

Wa(xi)va = W(xi)v (38)

in which Wa are test functions and va arbitrary parameters. Using this form ofapproximation will convert equation (34) to a set of algebraic equations.

In the finite element method and indeed in all other numerical procedures forwhich a computer-based solution can be used, the test and trial functions willgenerally be defined in a local manner. It is convenient to consider each of thetests and basis functions to be defined in partitions e of the total domain . Thisdivision is denoted by:

≈ h =⋃e (39)

and in a finite element method e are known as elements. The very simplest useslines in one dimension, triangles in two dimensions and tetrahedra in three dimen-sions in which the basis functions are usually linear polynomials in each elementand the unknown parameters are nodal values of φ. In Figure 4.25 we show a typicalset of such linear functions defined in two dimensions.

In a weighted residual procedure, we first insert the approximate function φinto the governing differential equation creating a residual, R(xi), which of courseshould be zero at the exact solution. In the present case for the quasi-harmonicequation we obtain:

R = − ∂

∂xi

(k∑

a

∂Na

∂xiφa

)+ Q (40)



a

y

x

Na∼

Figure 4.25. Basis function in linear polynomials for a patch of triangular elements.

and we now seek the best values of the parameter set φa, which ensures that:∫

WbRd = 0, b = 1, 2, . . . , n (41)

Note that this is the term multiplying the arbitrary parameter vb. As notedpreviously, integration by parts is used to avoid higher-order derivatives (i.e. thosegreater than or equal to two) and therefore reduce the constraints on choosing thebasis functions to permit integration over individual elements using equation (39).In the present case, for instance, the weighted residual after integration by partsand introducing the natural boundary condition becomes:

∫

∂Wb

∂xi

(k∑

a

∂Na

∂xiφ

)d+

∫

WbQd+∫q

Wbqnd = 0 (42)

4.2.3 The Galerkin, finite element, method

In the Galerkin method we simply take Wb = Nb, which gives the assembled systemof equations:

n∑a=1

Kbaφa + fb = 0, b = 1, 2, . . . , n − r (43)

where r is the number of nodes appearing in the approximation to the Dirich-let boundary condition (i.e., equation (30)) and Kba is assembled from elementcontributions Ke

ba with:

Keba =

∫e

∂Nb

∂xik∂Na

∂xid (44)



Similarly, fb is computed from the element as:

f eb =

∫e

NbQd+∫eq

Nbqnd (45)

To impose the Dirichlet boundary condition we replace φa by φa for the rboundary nodes.

It is evident in this example that the Galerkin method results in a symmetric set ofalgebraic equations (e.g. Kba = Kab). However, this only happens if the differentialequations are self-adjoint. Indeed the existence of symmetry provides a test forself-adjointness and also for existence of a variational principle whose stationarityis sought.

It is necessary to remark here that if we were considering a pure convectionequation:

ui∂φ

∂xi+ Q = 0 (46)

symmetry would not exist and such equations can often become unstable if theGalerkin method is used.

4.2.4 Characteristic Galerkin scheme for convection–diffusion equation

Unlike a simple conduction equation (as the Laplace equation), a numerical solutionfor the convection equation has to deal with the convection part of the governingequation in addition to diffusion. For most conduction equations, the finite elementsolution is straightforward. However, if a Galerkin type approximation was usedin the solution of convection equations, the results will be marked with spuriousoscillations in space if certain parameters exceed a critical value (element Pecletnumber). This problem is not unique to finite elements as all other spatial dis-cretization techniques have the same difficulties. A very well-known method usedin finite elements approximation to reduce these oscillations is the CharacteristicGalerkin (CG) scheme (Lewis et al. [19], Zienkiewicz et al. [20]). Here, we followthe Characteristic Galerkin (CG) approach to deal with spatial oscillations due tothe discretization of the convection transport terms.

In order to demonstrate the CG method, let us consider the simple convection–diffusion equation in one dimension, namely:

∂φ

∂t+ u1

∂φ

∂x1− ∂

∂x1

(k∂φ

∂x1

)= 0 (47)

Let us consider a characteristic of the flow as shown in Figure 4.26 in the time–space domain. The incremental time period covered by the flow ist from the nthtime level to the n + 1th time level and the incremental distance covered during this



Characteristic

n + 1

n

x1φ n + 1

x1φ n

∆t

x1− ∆x1 ∆x1x1

x1− ∆x1φ n

Figure 4.26. Characteristic in a space–time domain.

time period isx1, that is, from (x1 −x1) to x1. If a moving coordinate is assumedalong the path of the characteristic wave with a speed of u1, the convection termsof equation (47) disappear (as in a Lagrangian fluid dynamics approach). Althoughthis approach eliminates the convection term responsible for spatial oscillationwhen discretized in space, the complication of a moving coordinate system x′

1 isintroduced, that is, equation (47) becomes:

∂φ

∂t(x′

1, t) − ∂

∂x′1

(k∂φ

∂x′1

)= 0 (48)

The semi-discrete form of the above equation can be written as:

φn+1|x1

− φn|x1−x1

t− ∂

∂x′1

(k∂φ

∂x′1

)n

|x1−x1

= 0 (49)

Note that the diffusion term is treated explicitly. It is possible to solve the aboveequation by adapting a moving coordinate strategy. However, a simple spatialTaylorseries expansion in space avoids such a moving coordinate approach. With referenceto Figure 4.26, we can write using a Taylor series expansion:

φn|x1−x1= φn|x1

− ∂φn

∂x1

x1

1! + ∂2φn

∂x21

x21

2! − . . . (50)

Similarly, the diffusion term is expanded as:

∂

∂x′1

(k∂φ

∂x′1

)n

|x1−x1

= ∂

∂x1

(k∂φ

∂x1

)n

|x1

− ∂

∂x1

[∂

∂x1

(k∂φ

∂x1

)n]x (51)



ji

l

Figure 4.27. One-dimensional linear element.

On substituting equations (50) and (51) into equation (49), we obtain (higher-order terms being neglected) the following expression:

φn+1 − φn

t= −x

t

∂φ

∂x1

n

+ x2

2t

∂2φ

∂x21

n

+ ∂

∂x1

(k∂φ

∂x1

)n

(52)

In this case, all the terms are evaluated at position x1, and not at two positionsas in equation (49). If the flow velocity is u1, we can writex = u1t. Substitutinginto equation (52), we obtain the semi-discrete form as:

φn+1 − φn

t= −u1

∂φ

∂x1

n

+ u21t

2

∂2φ

∂x21

n

+ ∂

∂x1

(k∂φ

∂x1

)n

(53)

By carrying out a Taylor series expansion (Figure 4.26), the convection termreappears in the equation along with an additional second-order term. This second-order term acts as a smoothing operator that reduces the oscillations arising fromthe spatial discretization of the convection terms. The equation is now ready forspatial approximation.

The following linear spatial approximation of the scalar variable ϕ in space isused to approximate equation (53):

φ = Niφi + Njφj = [N]ϕ (54)

where [N] are the shape functions and subscripts i and j indicate the nodes of alinear element as shown in Figure 4.27.

On employing the Galerkin weighting to equation (53), we obtain:∫

[N]T φn+1 − φn

td+

∫

[N]T u1∂φ

∂x1

n

d

(55)

− t

2

∫

[N]T u21∂2φ

∂x21

n

d−∫

[N]T ∂

∂x1

(k∂φ

∂x1

)n

= 0

The above equation is equal to zero only if all the element contributions areassembled. For a domain with only one element, we can substitute:

[N]T =[

Ni

Nj

](56)



On substituting a linear spatial approximation for the variable φ, over elementsas typified in Figure 4.27, into equation (55), we get:∫

[N]T [N]φn+1 − φn

td = − u1

∫

[N]T ∂

∂x1([N]φ)nd

(57)

+ t

2u2

1

∫

[N]T ∂2

∂x21

([N]φ)nd+∫

[N]T k∂2

∂x21

([N]φ)nd

Before utilizing the linear integration formulae, we apply Green’s lemma to thesecond-order terms of equation (57), we obtain:∫


td = −u1

∫

[N]T ∂[N]

∂x1φnd

− t

2u2

1

∫

∂[N]T

∂x1

∂[N]

∂x1φnd+ t

2u2

1

∫

[N]T ∂[N]

∂x1φnn1d (58)

−∫

∂[N]T

∂x1k∂[N]

∂x1φnd+

∫

[N]T k∂[N]

∂x1φnn1d

where n1 and n2 are the direction cosines of the outward normal n, is the domainand is the domain boundary. The first-order convection term can be integratedeither directly or via Green’s lemma. Here, the convection term is integrated directlywithout applying Green’s lemma. However, integration of the first derivatives byparts is useful for problems in which the traction is prescribed. Using the integrationformulae (Lewis et al. [19]), it is possible to derive the element matrices for all theterms in equation (58). The term on the left-hand side for a single element is:

∫


td = l

6

[2 11 2

]⎧⎨⎩φn+1

i −φni

t

φn+1j −φn

j

t

⎫⎬⎭ = [Me]ϕt

(59)

where [Me] is the mass matrix for a single element. The above mass matrix for asingle element will have to be utilized in an assembly procedure for a fluid domaincontaining many elements.

In a similar fashion, all other terms can be integrated; for example, theconvection term is given by:

u1

∫

[N]T ∂[N]

∂x1φnd = u1

2

[−1 1−1 1

]φi

φj

n

= [Ce]ϕn (60)

where [Ce] is the elemental convection matrix. The values of the derivatives of theshape functions are substituted in order to derive the above matrix.



The diffusion term within the domain is integrated as:∫

∂[N]T

∂x1k∂[N]

∂x1φnd = k

l

[1 −1

−1 1

]φi

φj

n

= [Ke]ϕn (61)

where [Ke] is the elemental diffusion matrix. The characteristic Galerkin termwithin the domain is integrated as:

t

2u2

1

∫

∂[N]T

∂x1

∂[N]

∂x1φnd = u2

1t

2

1

l

[1 −1

−1 1

]φi

φj

n

= [Kse]ϕn (62)

where [Kse] is the elemental stabilization matrix.The boundary term from the diffusion operator is integrated by assuming that i

is a boundary node, as follows:

∫

[N]T k∂[N]

∂x1φnn1d = k

⎧⎨⎩−φi

l+ φj

l0

⎫⎬⎭n

n1 = [fe] (63)

where f e is the forcing vector due to the diffusion term.The boundary integral from the characteristic Galerkin term is integrated, again

by assuming that i is a boundary node, as:

t

2u2

1

∫

[N]T ∂[N]∂x1

φnn1d = t

2u2

1

⎧⎨⎩−φi

l+ φj

l0

⎫⎬⎭n

n1 = [fse] (64)

where f se is the forcing vector due to the stabilization term.For a one-dimensional domain with more than one element, all the matrices and

vectors need to be assembled in order to obtain the global matrices. Once assembled,the discretized one-dimensional equation becomes:

[M]ϕt

= −[C]φn − [K]φn − [Ks]φn + fn + fsn (65)

Let us now consider a simple one-dimensional convection problem, as given inFigure 4.28, to demonstrate the effect of a discretization with and without the CGscheme.

The scalar variable value at the inlet is ϕ= 0, and at the exit its value is 1. Thisscalar variable is transported in the direction of the velocity as shown in Figure4.28. Note that the convection velocity u1 is constant. The element Peclet numberfor this problem is defined as:

Pe = u1h

2k(66)

where h is the element size in the flow direction, which, in one dimension is thelocal element length. Figure 4.29 shows the comparison between a solution with



L

φ = 0Inlet

u1 = constant φ = 1Exit

Figure 4.28. One-dimensional convection–diffusion problems.

0.8

1

0.6

Fun

ctio

n

0.4

0.2

0

0 0.2 0.4Horizontal distance

(a) Pe = 1.0

0.6 0.8 1

0.8

1

0.6F

unct

ion

0.4

0.2

0

0 0.2 0.4Horizontal distance

(b) Pe = 1.5

0.6 0.8

Standard GalerkinCharacteristic Galerkin

Exact solution

1

Standard GalerkinCharacteristic Galerkin

Exact solution

Figure 4.29. Spatial variation of a function, φ, in one-dimensional space fordifferent element Peclet numbers.

the CG discretization scheme and one without it. Only two Peclet numbers areshown in these diagrams to demonstrate the spatial oscillations without the CGdiscretization. As seen, both discretizations give no spatial oscillations at a Pevalue of unity. However, at a Pe value of 1.5, the CG discretization is accurate andstable, while the discretization without the CG term becomes oscillatory. The exactsolution to this problem is given as follows (Brooks and Hughes [21]):

φ = 1 − eu1x1

k

1 − eu1L

k

(67)

In this equation, L is the total length of the domain and x1 is the local length ofthe domain.

The extension of the characteristic Galerkin scheme to a multi-dimensionalscalar convection–diffusion equation is straightforward and follows the previousprocedure as discussed for a one-dimensional case.

4.2.5 Stability conditions

The stability conditions for a given time discretization may be derived usinga Von Neumann or Fourier analysis for either the convection equation or theconvection–diffusion equations. However, for more complicated equations such



A3

l3

l2

l1

l4

l5

Node

A2

A1

A4

A5

Figure 4.30. Two-dimensional linear triangular element.

as the Navier–Stokes equations, the derivation of the stability limit is not straight-forward. A detailed discussion on stability criteria is not given here and readersare asked to refer to the relevant text books and papers for details (Hirsch [22],Zienkiewicz and Codina [23]). A stability analysis will give some idea about thetime-step restrictions of any numerical scheme. In general, for fluid dynamics prob-lems, the time-step magnitude is controlled by two wave speeds. The first one isdue to the convection velocity and the second due to the real diffusion introducedby the equations. In the case of a convection–diffusion equation, the convection

velocity is√

uiui, which is, in two dimensional problems,√

u21 + u2

2 = |u|. The dif-fusion velocity is 2k/h where h is the local element size. The time-step restrictionsare calculated as the ratio of the local element size and the local wave speed. It istherefore correct to write that the time step is calculated as:

t = min (tc,td ) (68)

where tc and td are the convection and diffusion time-step limits, respectively,which are:

tc = h

|u| ,td = h2

2k(69)

Often, it may be necessary to multiply the time-step t by a safety factor dueto different methods of element size calculations. A simple procedure to calculatethe element size in two dimensions is:

h = min(

2Areai

li

), i = 1, number of elements connected to the node (70)

where Areai are the area of the elements connected to the node and li are the lengthof the opposite sides as shown in Figure 4.30. For the node shown in this figure, thelocal element size is calculated as:

h = min (A1/l1, A2/l2, A3/l3, A4/l4, A5/l5) (71)



In three dimensions, the term 2Areai is replaced by 3Volumei and li is replacedby the area opposite the node in question.

4.2.6 Characteristic-based split scheme

It is essential to understand the characteristic Galerkin procedure, discussed inSection 2.4 for the convection–diffusion equation, in order to apply the concept tosolve the real convection equations. Unlike the convection–diffusion equation, themomentum equation, which is part of a set of heat convection equations, is a vectorequation. A direct extension of the CG scheme to solve the momentum equationis difficult. In order to apply the CG approach to the momentum equations, wehave to introduce two steps. In the first step, the pressure term from the momentumequation will be dropped and an intermediate velocity field will be calculated.In the second step, the intermediate velocities will be corrected. This two-stepprocedure for the treatment of the momentum equations has two advantages. Thefirst advantage is that without the pressure terms, each component of the momentumequation is similar to that of a convection–diffusion equation and the CG procedurecan be readily applied. The second advantage is that removing the pressure termfrom the momentum equations enhances the pressure stability and allows the useof arbitrary interpolation functions for both velocity and pressure. In other words,the well-known Babuska–Brezzi condition is satisfied (Babuska [24], Brezzi andFortin [25], Chung [26]). Owing to the split introduced in the equations, the methodis referred to as the CBS scheme.

The CG procedure may be applied to the individual momentum componentswithout removing the pressure term, provided the pressure term is treated as asource term. However, such a procedure will lose the advantages mentioned inthe previous paragraph. For more mathematical details, please refer to Zienkiewiczet al. [20], Zienkiewicz and Codina [23], Nithiarasu [27] and Zienkiewicz et al. [28].

In order to apply the CG procedure, we can refer to the general case of governinggeneralized porous medium flow and heat transfer equations in non-dimensionalform and indicial notation, for mixed convection, that have been presented inSection 1.3 (see equations (24)–(26)).

From the governing equations, it is obvious that the application of the CGscheme is not straightforward. However, by implementing the following procedure,it is possible to obtain a solution to the convection heat transfer porous mediumequations. The solution of the free fluid flow equations is obtained by applying thesame procedure.

Temporal discretizationFor the sake of simplicity, the asterisks are omitted and the Darcy and Forchheimerterms in equation (25) are grouped as a “porous” term to obtain:

P =(

1ReDa

+ F√Da

|u|)

(72)



Temporal discretization along the characteristics of continuity, momentum andenergy conservation equations results in the following set of equations:

∂un+ϑ1i

∂xi= 0 (73)

un+1i (1 +tεPϑ3) − un

i (1 −tεP(1 − ϑ3))

= −εt

(ϑ2

∂pn+1f

∂xi+ (1 − ϑ2)

∂pnf

∂xi

)−

[t

εuj∂ui

∂xj

]n

+[t

J

Re

∂2ui

∂x2i

]n

+ εt2

2

[1

ε2uk∂

∂xk

(uj∂ui

∂xj

)+ uk

∂

∂xk

∂pf

∂xi

]n

+ εtRa

Pr Re2 T nγi (74)

T n+1 − T n

t= −

[ui∂T

∂xi

]n

+[

1

σRe Pr

∂2T

∂x2i

]n

+ t

2σ

[uk∂

∂xk

(ui∂T

∂xi

)]n

(75)

In equations (74) and (75), additional stabilization terms appear naturally fromthe discretization along the characteristics (Zienkiewicz et al. [20]), while the char-acteristic parameters ϑ1, ϑ2 and ϑ3 (0.5 ≤ϑ1 ≤ 1 and 0 ≤ϑi ≤ 1, with i = 2, 3) aredefined according to the following:

∂un+ϑ1i

∂xi= ϑ1

∂un+1i

∂xi+ (1 − ϑ1)

∂uni

∂xi,

∂pn+ϑ2f

∂xi= ϑ2

∂pn+1f

∂xi+ (1 − ϑ2)

∂pnf

∂xi, (76)

[Pui]n+ϑ3 = ϑ3[Pui]n+1 + (1 − ϑ3)[Pui]n.

Different versions of the CBS scheme can be obtained depending on the valueof the above parameters. In particular an SI and an AC version of the CBS schemecan be obtained by varying the parameter ϑ2. For ϑ2 between 0.5 and 1, the SI-CBSis obtained, while for ϑ2 equal to 0, the AC-CBS scheme is derived. Moreover, forϑ3 between 0.5 and 1, an implicit treatment for the porous term is obtained, whilefor ϑ3 equal to 0, an explicit one is derived.

The splitting in the CBS scheme consists of solving the above equations in anumber of subsequent steps. In the first step, the pressure term is removed fromequation (74) and the intermediate velocity components ui are obtained from:

Step 1: Intermediate velocity calculation

ui(1 +tεPϑ3) − uni (1 +tεP (ϑ3 − 1)) = −

[t

εuj∂ui

∂xj

]n

+[t

J

Re

∂2ui

∂x2i

]n

+ εt2

2

[1

ε2uk∂

∂xk

(uj∂ui

∂xj

)]n

+ εtRa

Pr Re2 T nγi (77)



Removing the pressure term from the momentum equation, the pressure stabilityis enhanced and the use of arbitrary interpolation functions for both velocity andpressure is allowed. In other words, the well-known Babuska–Brezzi condition issatisfied (Babuska [24], Brezzi and Fortin [25], Chung [26]). The correct velocitiescan be determined, once the pressure field is known, using the equation:

Step 3: Velocity correction

un+1i − ui = − 1( 1

εt + Pϑ3)[ϑ2

∂pn+1f

∂xi+ (1 − ϑ2)

∂pitart

∂xi+

(t

2uk∂

∂xk

∂pf

∂xi

)n]

(78)

The solution of equation (78) is the third step of the algorithm.The second step consists in the pressure calculation through the continuity equa-

tion. This second step is where the SI and AC versions of the scheme differ. Inparticular, this second step is obtained in the SI version of the scheme by derivingequation (78) with respect to xi and imposing equation (73), obtaining the followingPoisson type of equation:

Step 2 SI-CBS: Pressure calculation

0 = −t1

ε

(ϑ1∂ui

∂xi+ (1 − ϑ1)

∂uni

∂xi

)+t2

∂2pn+1f

∂x2i

(79)

Therefore, in this case, the incompressibility constraint is satisfied at each iter-ation, and the pressure, evaluated through equation (79), represents the actualpressure. However, the solution of equation (79) needs a matrix inversion. Further-more, the SI version of the scheme uses a global time step, which is the minimumvalue of the time step limit over the entire domain.

Alternatively, the AC scheme can be derived when the left hand side of equa-tion (79) is obtained from a mass conservation equation retaining the transientdensity term:

∂ρf

∂t+ 1

ε

∂(ρf ui)

∂xi= 0 (80)

In general, it is possible to relate the density time variation to the pressure timevariation, through the speed of sound, as follows:

∂ρf

∂t= 1

c2

∂pf

∂t(81)

where the real compressibility parameter, c (compressible wave speed), approachesinfinity for many incompressible flow problems and the solution scheme becomesstiff and imposes severe time step restrictions. However, this parameter can bereplaced locally by an appropriate artificial value, β, of finite value, employing the



AC method, that was first introduced by Chorin [29] and then further developed[30–42]:

∂ρf

∂t= 1

β2

∂part

∂t⇒ 1

β2

pit+1art − pit

art

t= −ρf

ε

∂uit+ϑ1i

∂xi(82)

In equation (82), the pressure is an artificial pressure, because the incompressibilityconstraint is not achieved at each time step, but only when steady-state convergenceis reached. The superscripts it + 1 and it are referred to the iterative procedure anddo not refer to real time levels. The second step of the AC scheme is carried out byimposing the equation (82) as:

Step 2 AC-CBS: Pressure calculation

1

β2 (pit+1art − pit

art) = −t1

ε

(ϑ1∂ui

∂xi+ (1 − ϑ1)

∂uiti

∂xi

)+t2 ∂

2pitart

∂x2i

(83)

The local value of β is calculated through the following procedure:

β = max (0.5, uconv, udiff , uther) (84)

The local convective, diffusive and thermal velocities can be calculated throughthe following non-dimensional relations:

uconv = √uiui , udiff = 2

hRe, uther = 2

hRe Pr. (85)

The diffusive time step limitation for the AC method may be written as h2/Re/2,while the convective time step limitation may be written as:

tconv = h

|u| + β (86)

The above relation includes the viscous effect via the artificial parameter.Depending on the problem of interest, it is possible to consider other equations

coupled to the above set, such as species concentration for multi-component flows.The fourth step, for non-isothermal problems, is represented by the energy con-

servation equation (75) that allows calculating the temperature at every iteration forproblems with a coupling between momentum and energy conservation equationsoccurs.

Spatial discretization and solution methodologyThe spatial discretization of the governing equations is obtained through Galerkinfinite element procedure and triangular elements. Within an element, each variableis calculated through linear approximation on the basis of nodal values, accordingto the following equation:

φ =3∑

n=1

Nnφn (87)



where Nn is the shape function at node n and φn is the value of the generic variableφ at node n. Applying the standard Galerkin procedure to the set of equationspresented in the previous section, the following four steps, expressed in a matrixform, are obtained:

Step 1: Intermediate velocity calculation

Mui = (1 −tεP)Muni − t

ε[Cui]n −t

J

Re[Kdui]n − 1

2

t2

ε[Kuui]n

(88)

+εt

[Ra

Pr Re2 MTγi

]n

+tJ

Re[fd]n + 1

2

t2

ε[fu]n

Step 2 SI-CBS: Pressure calculation

[Kppf ]n+1 = − 1

εt[Dui] (89)

Step 2 AC-CBS: Pressure calculation

1β2 M(pit+1

art − pitart) = −t

ε[Dui] −t2[Kppart]it (90)

Step 3: Velocity correction

M(un+1i − ui) = −εt

[ϑ2Dpn+1

f + (1 − ϑ2)Dpitart − t

2Kupn

f

](91)

Step 4: Temperature calculation

MT n+1 = MT n −t[CT ]n − tλ

Re Pr Rk[KdT ]n

(92)

− t2

2Rk[KuT ]n + tλ

Re Pr Rk[ft]

n + t2

2Rk[fut]

n

where M is the mass matrix, C is the convection matrix, Kd is the diffusion matrix,Ku is the stabilization matrix obtained from higher-order terms, fd and fu are theboundary vectors from the momentum equation, Kp is the stiffness matrix, D is thegradient matrix, f t and fut are the boundary vectors from the temperature equation.Details of all the terms presented in equations (88)–(92) are given in Lewis et al.[19] and Zienkiewicz et al. [20].

In the present procedure, the mass matrix is lumped using a standard row-summing approach, inverted and stored in an array during the pre-processing.Therefore, in the AC formulation, the calculation of the artificial pressure at it + 1iteration, through equation (90), does not need a matrix inversion and the result isa matrix inversion-free procedure. On the other hand, in the second step of the SIsolution procedure, the stiffness matrix needs to be inverted at each iteration.



u = 0, v = 0, q = 1

u = 0, v = 0, q = 1

u = 1T = 0

y

x

(b)

(a)

Figure 4.31. Forced convection in a porous channel. (a) Computational domain andboundary condition; (b) structured computational grid (3,321 nodes,6,400 elements).

Some examplesBoth the schemes presented above can be used to solve many engineering problems,such as mixed, natural or forced convection both in fluid-saturated porous mediaand in partly porous domains, or simply in free fluid flow cases.

As mentioned before, changing the properties such as porosity and thus per-meability, it is possible to handle porous medium-free fluid interface problems asa single problem with different properties. The following limits are used for theporous medium part and for the free fluid part:

ε < 1Da = finite

⇒ porous mediumε = 1

Da → ∞ ⇒ free fluid (93)

A suitable set of matching conditions, to connect the porous medium and thefree fluid domains, is needed (Massarotti et al. [43]).

The first example is the forced convection heat transfer inside a uniform porouschannel with constant wall heat flux. The computational domain, together with theboundary conditions employed, is sketched in Figure 4.31a, while Figure 4.31bshows the computational grid employed. The flow enters the domain from the leftwith a constant velocity and at a non-dimensional temperature T = 0. The channelwalls are heated by a constant heat flux. The numerical results have been obtainedusing a non-uniform grid with 6,400 elements and 3,321 nodes.

Figure 4.32 shows the non-dimensional velocity and temperature profiles, fordifferent Darcy numbers, varying from 10−4 to 1.0 at a section, where the flowand the thermal field are hydrodynamically and thermally developed. The resultshave been compared with the analytical solutions presented by Nield et al. [44] andLauriat and Vafai [45]. The dimensionless temperature is evaluated using:

Tmix = 1umeanSx

∫Sx

uiTidS (94)



y

u/u

mea

n

−1 −0.5 0

(a) (b)

0.5 10

0.5

1

1.5

Analytical Nieldet al. [44]AC-CBSSI-CBS

Da = 10−4

Da = 10−2Da = 5x10−2 Da =10−1

Da = 1

yT

T

wal

l/T

mix

T

wal

l

−1 −0.5 0 0.5 10

0.5

1

1.5

Analytical Lauriatand Vafai [45]AC-CBSSI-CBS

Da = 10−3

Da = 10−1

Da = 1

Da = 10−4

Figure 4.32. Forced convection in a porous channel: (a) non-dimensional velocityas function of transverse distance; (b) non-dimensional temperatureas function of transverse distance.

where umean is the mean velocity in the section considered Sx.For a small Darcy number (10−4), the velocity profile is nearly independent of

the transverse distance and slip flow occurs at the walls (Figure 4.32). An increasein Darcy number leads to a non-linear distribution of the velocity. When the Darcynumber is equal to unity, the velocity profile approaches a profile similar to that offree fluid flow.

The non-dimensional temperature increases as the Darcy number decreases, asshown in Figure 4.32. A decrease in the Darcy number corresponds to a decreasein the fluid velocity in the middle of the channel and therefore the conduction heattransfer is dominating over the effect of convection.

Figure 4.33 shows the convergence histories to steady state for the AC-CBS andSI-CBS schemes obtained by using the following L2 norm for velocities:

Velocity residual =√√√√nodes∑

i=1

( |ui|n+1 − |ui|nti

)2/

nodes∑i=1

( |ui|n+1

ti

)2

(95)

This figure shows that the SI version of the scheme converges faster as the Darcynumber decreases. Instead, theAC version of the scheme has an opposite behaviour.Table 4.1 reports the CPU time needed by both the procedures to reach steady statesolution, on a machine with 4 Gb of RAM and 2.4 GHz CPU speed. These resultsconfirm the behaviour shown in Figure 4.33. In particular, Table 4.1 shows thatthe AC-CBS scheme converges faster than the SI-CBS scheme when Da ≥ 10−2. Itwas noticed that the SI-CBS scheme needs more time than the AC-CBS scheme pertime iteration. In correspondence of smaller Darcy numbers (Da ≤ 10−3), the AC-CBS scheme takes more time to reach the steady state. This is due to the time step



Iterations

Velo

city

resi

dual

100

100 101 102 103 104 105

10−1

10−2

10−3

10−4

10−5

10−6

AC-CBSSI-CBS

(a)Iterations

Velo

city

resi

dual

100

100 101 102 103 104 105

10−1

10−2

10−3

10−4

10−5

10−6

AC-CBSSI-CBS

(b)

Figure 4.33. Convergence histories for SI and AC scheme. (a) Da = 1;(b) Da = 10−4.

Table 4.1. CPU time (s) for forced convection in a porous channel at different Darcynumbers

Da 1 10−1 5 × 10−2 10−2 10−3 10−4

AC-CBS 35 87 137 491 4,341 6,400

SI-CBS 14,730 10,250 8,055 3,357 1,107 250

calculation procedure used. Essentially, theAC scheme becomes slower in reachingthe steady state when diffusion is overriding the effect of convection (Da ≤ 10−3)and the local time step approaches the global time step, losing the advantage ofusing a higher time step in the convective zones.

The second example is the natural convection in a cavity heated uniformly fromthe bottom side. The computational domain, together with the boundary conditionsemployed, is sketched in Figure 4.34a, while Figure 4.34b shows the computationalgrid employed, composed of 7,200 triangular elements and 3,721 nodes.

Figure 4.35 shows the dimensionless temperature contours for a Prandtl numberequal to 0.71, and for two different Rayleigh numbers (106 and 7 × 104) and twodifferent Darcy numbers (10−3 and 10−4). The results obtained have been comparedwith the numerical solution presented by Basak et al. [46].

In general, the fluid circulation is strongly dependent on the Darcy number.In fact, when smaller Rayleigh and Darcy numbers are considered, the flow isvery weak and the temperature distribution is similar to that of stationary fluid(Figure 4.35a). As the Darcy number increases, the role of convection becomes



1

(a) (b)

u=v=0

T=0

u=v=0, T=0

u=v=0

y

xT=0

u=v=0, =0∂T

∂y

Figure 4.34. Natural convection in a porous cavity heated from the bottom side:(a) computational domain and boundary conditions; (b) structuredcomputational grid (3,721 nodes, 7,200 elements).

0.4

0.1

0.4

0.3

0.1

0.30.1

0.1

0.80.70.5

0.4

0.1

0.3

0.2

0.90.8

0.7

0.50.4

0.6

0.3

0.1

0.1 0.1

0.40.20.3

0.5

0.1

0.40.30.2

0.1

0.40.9

0.2

0.2

0.1

0.6

0.70.2 0.6 0.5

0.8 0.v 0.60.8 0.7

0.2

0.30.40.5

0.7

0.6

0.40.5

0.5

0.5

0.4 0.

3

0.3

0.4

0.3

0.9

0.5

0.2

0.2

0.5

0.2

(a) (b)

Figure 4.35. Temperature contours for natural convection in a porous cavity heatedfrom the bottom side. (a) Ra = 7 × 104 and Da = 10−4; (b) Ra = 106

and Da = 10−3.

more significant and the fluid rises up strongly from the middle portion of thebottom wall, as depicted in Figure 4.35b.

Figure 4.36 shows the steady-state convergence histories for the AC-CBS andthe SI-CBS schemes. This figure shows a different behaviour from that of the caseof forced convection flow problem. When natural convection occurs, the AC-CBSscheme converges faster than the SI-CBS scheme for any Rayleigh or Darcy number,both in terms of number of iterations (Figure 4.36) and CPU time to reach the steadystate (Table 4.2).

The SI-CBS algorithm takes more time to reach the steady state because of thecoupling between momentum and energy conservation equations present in naturalconvection problems. The coupling is due to the presence of a generation term on


166 Computational Fluid Dynamics and Heat TransferV

eloc

ity r

esid

ual

10−6

10−5

105

10−4

10−2

10−3

103

10−1

101

100

Vel

ocity

res

idua

l

10−6

10−5

10−4

104

10−2

102

10−3

10−1

100

100

Iterations105103101 104102100

Iterations

AC-CBSSI-CBS

AC-CBSSI-CBS

(a) (b)

Figure 4.36. Convergence histories for SI andAC. (a) Ra = 7 × 104 and Da = 10−4;(b) Ra = 106 and Da = 10−3.

Table 4.2. CPU time (s) for natural convection in a porous cavity heated from thebottom at different Rayleigh and Darcy numbers

Ra 106 7 × 104

Da 10−3 10−4 10−3 10−4

AC-CBS 180 930 290 700

SI-CBS 63,270 48,230 53,400 47,560

the right hand side of the momentum conservation equation in gravity directionthat must be evaluated at each time step. This term can be calculated only after theresolution of the energy equation (step 4 of the CBS algorithm), due to the fact thatthe nodal temperature values are unknown. This coupled system becomes stiff andcauses a restriction on the global time step value used in the SI procedure and thusthe solution to the simultaneous equations at the second step of the algorithm needsmore time.

Moreover, the SI scheme experiences difficulties in reaching the steady statewhen very refined meshes are used (Massarotti et al. [47]). On the other hand, theAC-CBS scheme does not show any problem in reaching the convergence whennatural convection flow occurs. The reason for this is the robust local time steppingprocedure used by the AC-CBS scheme.

The last example is the developing mixed convection in a region partially filledwith a fluid-saturated porous medium, confined between two vertical hot walls.In particular, because of the geometrical symmetry, half domain has been studied.Figure 4.37 shows the computational domain and the boundary conditions employedand the details of the computational grid near the entrance. The mesh employed has



u = v=0

(a) (b)

T = 1

v = 1, T = 0

L

yx

u = v=0T = 1

Figure 4.37. Mixed convection in a vertical channel partially filled with aporous medium. (a) Computational domain and boundary conditions;(b) detail of the structured computational grid near the entrance (8,591nodes and 16,800 elements).

x0 1

0

1

2

v/v

0

y = 3

y = 32 y = 0.5

Chang [48]

AC-CBS

1.5

0.5

0.5

Figure 4.38. Mixed convection in a vertical parallel plate channel partially filledwith a porous medium. Non-dimensional vertical velocity at differentheights of the channel and at Da = 10−5.

8,591 nodes and 16,800 elements and is refined near the wall, near the inlet regionand at the interface.

Figure 4.38 shows the dimensionless vertical velocity profile at different heightsof the channel (y) for a Darcy number equal to 10−5, obtained by using theAC-CBSscheme. The present results are compared with the numerical results of Chang and



Chang [48]. The parameters used for the present investigation are Ra = 0.72 × 103,Pr = 0.72, Re = 50, λ= 2.8 in the porous region and λ= 1 in the free fluid region,ε= 0.8. When y is small, there is a large velocity difference at the interface of thecomposite system (Figure 4.38). When y increases, the flow discharge in the porouslayer decreases, and the peak of the fluid velocity profile moves to the central axisof the vertical parallel-plate channel.

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[42] Wasfy, T., West, A. C., and Modi, V. Parallel finite element computation of unsteadyincompressible flows. International Journal of Numerical Methods in Fluids, 26,pp. 17–37, 1998.

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[44] Nield, D. A., Kuznetsov, A. V., and Xiong, M. Effect of local thermal non-equilibriumon thermally developing forced convection in a porous medium. International Journalof Heat Mass Transfer, 45, pp. 4949–4955, 2002.

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[48] Chang, W.-J., and Chang, W.-L. Mixed convection in a vertical parallel-plate channelpartially filled with porous media of high permeability. International Journal of Heatand Mass Transfer, 39(7), pp. 1331–1342, 1996.


5 Equal-order segregated finite-elementmethod for fluid flow and heat transfersimulation

Jianhui XieMFG-Engineering Design & Simulation, Autodesk, Inc.,San Rafael, CA, USA

Abstract

This chapter introduces finite element–based equal-order mixed-GLS (GalerkinLeast Squares) segregated formulation for fluid flow and heat transfer. This schemefollows the idea used in finite-volume method, which decouples the fluid pressurecalculation and velocity calculation by taking the divergence of the vector momen-tum equation and applying some clear insights regarding incompressible flow. Theinherent velocity correction step in “segregated scheme” makes it attractive forsolution robustness and much faster compared to original fluid velocity–pressurecoupling scheme because mixed-GLS finite-element formulation does not assurethe mass conservation between the interface of the elements in comparison to finite-volume method. With these advantages, this formulation has prevailed in recentyears in major finite element–based commercial codes used in the industry sim-ulation models that usually import geometry from CAD and have large scale ofdegree of freedoms to solve. Taking example of fluid–thermal coupling problem,this chapter also discusses the strategies used to solve multiphysics problem in non-linear iteration level and the block I/O technique that uses finite-element methodfor handling massive data. Finally, this chapter illustrates some typical industryconjugate heat transfer problems where both solid and fluid parts are involved.

Keywords: Finite elements, Fluid flow, Heat transfer, Segregated formulation

5.1 Introduction

The quest for robust and efficient strategies to solve large, sparse matrix sys-tems has been an active area of research and development for decades. Galerkinfinite-element methods, with the advantages of handling irregular geometry andmultiphysics coupling, have played an important role in computational fluid dynam-ics area, but its computational difficulties and perceived shortcomings have led tothe development of alternative finite-element models.



Theoretically, the efforts have been to avoid the two major problems encoun-tered. The first problem is the “wiggles” velocity field problem related to thenode-to-node oscillation of velocity components emanating from boundaries withlarge velocity gradient. This problem is mostly pronounced under the condition ofhigh Reynolds numbers and coarse computational mesh, and therefore the inabil-ity of the finite-element mesh to handle the steep gradient results in an imbalancebetween advective and diffusive contribution. With getting best fit to the prob-lem, the weighted residual formulation produces a field that oscillates about thetrue solution (wiggle pattern). The second problem is the “saddle-point” prob-lem originated by mixed Galerkin finite-element formulation. It is difficult tosolve especially in three-dimensional large-scale industry model because the LBB(Ladyzhenskaya–Babuska–Brezzi) stability condition associated with the Galerkinfinite-element limits element choice with respect to the velocity and pressureinterpolation.

In practical aspect, solution schemes/solvers and algorithms have been exploredfor better CFD processor efficiency and robustness, which are mostly addressed bycommercial codes.

In this chapter, the equal-order mixed-GLS (Galerkin Least Squares) stabi-lized formulation is presented first. It is termed as a residual method becausethe added least-squares terms are weighted residuals of the momentum equa-tion, and this form of the least-squares terms implies the consistency of themethod since the momentum residual is employed. And this method is a gen-eralization of SUPG (streamline-upwind/Petrov–Galerkin) and PSPG (pressurestabilizing/Petrov–Galerkin) methods, which were developed to target the wig-gle stabilization. Since this method plays a significant stabilization role for thecoarse mesh and hybrid mesh, the equal-order velocity and pressure interpolationis viable, and the LBB condition is not necessary any more, the convenient bilinearelement interpolation becomes practical approximation although it was unstable inthe Galerkin finite-element context.

The second emphasis in this chapter is on the practical view: The numer-ical strategies for the solution of large systems of equations arising from thefinite-element discretization of the above formulations are discussed. To solve thenonlinear fluid flow and heat transfer problem, particular emphasis is placed on seg-regated scheme (which uses SIMPLE algorithm from the finite-volume method) innonlinear level and iterative methods in linear level. The velocity–pressure cou-pling formulation not only results in a large dimensioned system but also generatesa stiffness matrix radically different from the narrowband type of matrix, which isinefficient to be solved. Segregated scheme was proposed with the idea of decou-pling the pressure calculation and velocity calculation by taking the divergenceof the vector momentum equation and applying some clear insights regardingincompressible flow. The inherent velocity correction step in “segregated scheme”makes it attractive for solution robustness and faster compared to velocity–pressurecoupling scheme because mixed-GLS finite-element formulation does not assurethe mass conservation between the interface of the elements in comparison tofinite-volume method.


Equal-order segregated finite-element method 173

According to how the primitive variables of velocity–pressure are treated, finite-element methods for solving Navier–Stokes equations can be categorized into threegroups: (a) the velocity–pressure integrated method; (b) the penalty method; and(c) the segregated velocity–pressure method. The velocity–pressure integratedmethod in which the governing equations are treated simultaneously needs rela-tively small number of iterations to achieve convergence, but a large memory andcomputing time. The penalty method requires less memory and less computing timethan the first method, but it needs an additional postprocessor to obtain pressure fieldand satisfies the continuity equation only approximately. Recently, much attentionhas been paid to the segregated velocity–pressure method, where velocity and thecorresponding pressure field are computed alternatively in an iterative sequence.This method takes advantage of SIMPLE algorithm idea from finite-volume worldand uses it in the finite-element world. With the SIMPLE algorithm idea, it needsmuch less memory and execution time, and with the velocity correction step usingcontinuity equation, it even satisfies the continuity equation better than the firstmethod.

The second categorization of finite-element methods can be made accordingto the orders of interpolation functions for the velocity and pressure. They aremixed-order interpolation and equal-order interpolation. The mixed-order schemeattempts to eliminate the tendency to produce checkerboard pressure distributionsby satisfying LBB condition. In this method the velocity is interpolated linearlywhereas the pressure is assumed to be constant within the element. Equal-orderinterpolation was proposed by Rice and Schnipke [6]. They showed that equal-order scheme performed better for that purpose without exhibiting spurious pressuremodes.

A third categorization of the finite-element method for fluid dynamics isrelated to weight functions. Galerkin method has been widely used in discretiz-ing the momentum equation. Brook and Hughes demonstrated the accuracy ofthe streamline-upwind/Petrov–Galerkin method for the linear advection-diffusionequation; Rice and Schnipke [5] adopted a monotone streamline-upwind finite-element method, where they discretized the momentum equation by using con-vectional Galerkin method with the exception of the advection items, which weretreated by the monotone streamline-upwind approach.

Followed by the success of the finite-volume methods, several finite-elementsegregated solution schemes have been proposed. Rice and Schnipke [6] employedequal-order interpolation for all variables and solved pressure directly; however,their original formulation used a streamline unwinding scheme that may not bestraightforward enough to extend to three-dimensional flows. Early equal-ordermethods were in general of transient kind and the equations had to be integrated ontime to reach a steady state.Van Zijl [11] adapted the formulation by applying SUPGweighting functions to the convective terms; he later adapted the scheme furtherby implementing SIMPLEST algorithm. Shaw [7] demonstrated the way to useelement matrix construction for equal-order interpolation in segregated scheme.Haroutunian et al. [2] have shown that the implementation of iterative solverscan result in a substantial reduction in the storage requirements and execution



time. Du Toit [10] tried conjugate gradient solver; Wang [13,14] demonstrated a“consistent equal-order discretization method” by introducing element-based nodevelocity to satisfy mass conservation, while keeping conventional node velocityand temperature to satisfy momentum and energy equations. Wansophark [15]evaluated segregated schemes by combining monotone streamline-upwind methodand adaptive meshing technique. However, relaxation factors, which need user’sinterpretation, have remained an issue that affects the users.

The classical mixed velocity–pressure interpolation Galerkin finite-elementmethod, often referred to as “mixed v–p form,” was the workhorse of incom-pressible flow solvers in the 1970s to 1990s and has proved highly successful fortwo-dimensional and small three-dimensional problems, offering high order ofaccuracy and strong convergence rates [5]. The high mark of this approach is thediscretization of the continuity equation in a manner adherent to certain mathe-matical constraints, and the ability to simultaneously solve the equation with thediscretized momentum equations in either a Picard or a Newton–Raphson iterativescheme. The discretized continuity equation is the known cause of “indefiniteness”in the resulting Jacobian matrix, leading to unbounded large and negative eigenval-ues. Despite this poor matrix conditioning, most two-dimensional problems withsmall bandwidth are amenable to efficient solution with optimized variations ofclassical Gaussian elimination. For example, skyline method allows for memory-efficient solution and in fact is still used today. But even tremendous technologicaladvances in computer processor speed and random access memory capacity willnever make direct methods a viable alternative for large three-dimensional prob-lems. This is one reason why many researchers have abandoned the traditionalmixed-interpolation approach in favor of alternatives that allow the ready use of iter-ative solvers. These alternative formulations, including penalty methods, pressurestabilization methods, and pressure projection methods [2], bring “definiteness” tothe matrix system.

Transient analysis is a traditional way to overcome poor performance of iterativesolvers [9]; the idea is to take advantage of iterative solvers that thrive on a goodinitial guess – which transient analysis with small-enough time increment delivers ateach time step. Furthermore, the smaller the time step the more diagonally dominantthe matrix system, because the time derivatives occur on or near the diagonals. Itis noticed that the transient analysis is inefficient for a broad class of problemsthat want steady solution; hence, the result from integrating in time is not reallypractical as an end-all cure, instead transient term like inertial relaxation is moreeffective in steady analysis.

In general, finite-element method based simulation has following basic steps:

Preprocessor phase: The preprocessor phase prepares all the required data for thecore processor computation; it includes CAD input, mesh generation, and geom-etry decoder. After three-dimensional modeling from CAD packages, the modelgeometry and part/assembly relations are established in parameter control.

The CAD input module reads and transfers the CAD data into the mesh readygeometry data with the ordered hierarchy of assembly/part/surface relations.



Mesh generation, which is responsible for generating valid finite-elementmeshes and possible special meshes (e.g., boundary mesh for fluid flow), and meshrefinement.

Decoder does the job of producing final working finite-element nodes, elementconnectivity, node merge among parts, mesh clean up, contact, and setting up part,surface, node-based loadings with priority rules.

Processor phase: Processor phase is the kernel part of the finite-element application;it targets on solving real physical problems which are in discretized partial differ-ential equation (PDE) forms and nonlinear formulations. The processor takes careof several levels of iterations, which includes time iteration for transient problems;nonlinear iteration in a single time step handles the coupling among multiphysicsvariables like velocity, pressure, temperature, turbulence, and multiphase; theiterative solver for linear equations requires iteration for the converged solution.

In the inner level, element matrices are built up via shape function of the specificelement type and system matrix is assembled from the element-level matrix andthen eventually global linear algebraic equations are solved to produce the primaryresult.

The element-based and node-based results are the output from the processor;typical variables are velocity, pressure, temperature, turbulence kinetic energy anddissipation rate, phase fraction, etc.

The processor phase is usually the biggest user of system memory and CPUtime for finite-element-based simulation.

Postprocessor phase: Postprocessor also refers to result environment, whichprovides interactive graphics environment to visually interpret the node- andelement-based simulation results from processor phase. Some typical operationincludes slice plane, streamline, isoline and isosurface, make animation, annotation,derived date calculation, inquire data, curve plot and report production.

5.2 Finite-Element Description

A finite-element simulation program should select particular elements to form thebase element library. Elements for fluid flow and heat transfer analyses are usuallycategorized by the combination of velocity and pressure approximations used inthe element.

In finite element–based fluid flow and thermal problems, which is basicallyadvection–diffusion analysis, following elements are used for two-dimensional andthree-dimensional models.

5.2.1 Two-dimensional elements

Quadrilateral elementThe four-node quadrilateral element can be used to model either two-dimensionalCartesian or axisymmetric geometries. Each node has four degrees of freedom for



r

s 4 3

1 2

s 1

s 1

r 1r 1

Figure 5.1. Two-dimensional quadrilateral element.

laminar flow: U, V, P, T and six degrees of freedom for turbulent flows: U, V, P,T, K, ω. Quadrilateral element should have nine nodes if midside nodes are used(Figure 5.1).

Take this element as an example for using shape function and Jacobian matrix,the velocity component ui and temperature T are approximated by using bilinearinterpolation function,

ϕ = ϑ =

⎡⎢⎢⎢⎢⎣14 (1 − r)(1 − s)14 (1 + r)(1 − s)14 (1 + r)(1 + s)14 (1 − r)(1 − s)

⎤⎥⎥⎥⎥⎦ (1)

Two pressure discretizations are possible in this kind of element: a bilinear con-tinuous approximation, ψ ∈ Q1, with the pressure degrees of freedom located at thefour corner nodes, or a piecewise constant discontinuous pressure approximation,ψ= 1 ∈ Q0, with the pressure degrees of freedom associated with element centroid.

The construction of the finite-element matrices requires the computation ofvarious derivatives and integrals of the bilinear interpolation function. Since thebasis functions are given in terms of the normalized coordinates r and s and thederivatives and integrals are in terms of the physical x, y coordinates, the followingrelations need to be defined:⎡⎢⎢⎣

∂ϕ

∂r∂ϕ

∂s

⎤⎥⎥⎦ =

⎡⎢⎢⎣∂x

∂r

∂y

∂r∂x

∂s

∂y

∂s

⎤⎥⎥⎦⎡⎢⎢⎣∂ϕ

∂x∂ϕ

∂y

⎤⎥⎥⎦ =

⎡⎢⎢⎣∂N T

∂rx∂N T

∂sy

∂N T

∂sx∂N T

∂sy

⎤⎥⎥⎦⎡⎢⎢⎣∂ϕ

∂x∂ϕ

∂y

⎤⎥⎥⎦ = J

⎡⎢⎢⎣∂ϕ

∂x∂ϕ

∂y

⎤⎥⎥⎦ (2)

where J is the Jacobian matrix; by inverting it, equation (2) provides the necessaryrelation for the derivatives of the basis function as below:⎡⎢⎢⎣

∂ϕ

∂x∂ϕ

∂y

⎤⎥⎥⎦ = J −1

⎡⎢⎢⎣∂ϕ

∂r∂ϕ

∂s

⎤⎥⎥⎦ (3)



In the case of integral evaluation, to complete the transformation from physicalcoordinate to normalized coordinates, an elemental area for quadrilateral elementcould be evaluated as follows:

dx dy = |J |dr ds (4)

where |J | = Determine of J .Equations (3) and (4) allow any integral of functions of x and y to be expressed

as integrals of rational functions in the r and s coordinate system. This is extremelyimportant in evaluating element matrices, which is the basic step in finite-elementmethod.

The details for other elements are ignored in this chapter.

Triangular elementThe three-node triangular element can be used to model either two-dimensionalCartesian or axisymmetric geometries. Each node has four degrees of freedomfor laminar flow: U, V, P, T and six degrees of freedom for turbulent flows: U, V,P, T, K, ω.

Triangular element should have six nodes if midside nodes are used.

5.2.2 Three-dimensional elements

Brick (hexahedral) elementThe eight-node brick element is used to model three-dimensional geometries as thegeneral base. Each node has five degrees of freedom for laminar flow, i.e., V, W, P,T and six degrees of freedom for turbulent flows, i.e., V, W, P, T, K, ω.

Brick element has total 21 nodes if midside nodes are used; this is also used asthe base element for other degenerated elements with midside nodes.

Tetrahedral elementThe four-node tetrahedral element can be used to model three-dimensional geome-tries as a degenerated form from three-dimensional brick element; the mappingrelation is listed in Table 5.1. Each node has five degrees of freedom for laminarflow, i.e., U, V, W, P, T and seven degrees of freedom for turbulent flows, i.e., U, V,W, P, T, K, ω.

Tetrahedral element has 10 nodes if midside nodes are used.

Pyramid elementThe five-node pyramid element can be used to model three-dimensional geometriesas a degenerated form from three-dimensional brick element; the mapping relationis listed inTable 5.1. Each node has five degrees of freedom for laminar flow, i.e., U,V, W, P, T and seven degrees of freedom for turbulent flows, i.e., U, V, W, P, T, K, ω.

Pyramid element has 13 nodes if midside nodes are used.



Table 5.1. Mapping from degenerated three-dimensional elementsto computational nodal configuration

3

1 2

4 5

1 2

33

4 5

66

1 2

3

4

1 2

3

4 5

6

1 2

33

4 5

66

1 2

33

4 5

33

Degenerated elements Computational nodal configuration3D tetrahedral element

1 2 3 4 5 6 7 8

1 2 3 3 4 4 3 3

1 2 3 4 5 6 7 8

1 2 3 3 4 5 3 3

1 2 3 4 5 6 7 8

1 2 3 3 4 5 3 3

3D pyramid element

3D wedge element

Wedge elementThe six-node wedge (prism) element can be used to model three-dimensionalgeometries as a degenerated form from three-dimensional brick element; the map-ping relation is listed inTable 5.1. Each node has five degrees of freedom for laminar



flow, i.e., U, V, W, P, T and seven degrees of freedom for turbulent flows, i.e., U, V,W, P, T, K, .

Wedge element has 15 nodes if midside nodes are used.

5.2.3 Degenerated elements

In general finite-element code development, the three-dimensional six-nodewedges, five-node pyramids, and four-node tetrahedral are considered as degen-erate form of eight-node brick element by collapsing nodes. Table 5.1 attempts todemonstrate how degenerate elements are formed by collapsing nodes if midsidenodes are not used. The first row is node ID in brick element and the second row isfor degenerated nodes; the two rows make a mapping between degenerated elementnodes and brick element nodes. The high-order elements with midside nodes arenot commonly used for finite element–based fluid flow and heat transfer codes, andthis is discussed in later sections of this chapter.

In two-dimensional case, the three-node triangle element is degenerated fromfour-node quadrilateral element in a similar manner.

5.2.4 Special elements (rod and shell)

In heat transfer of some finite-element codes, arbitrary thickness rod and shellelements may be defined as those which allow heat conduction along the rod or inthe shell of the element, but offer no resistance to heat flow across the element.

Rod elements have only length and may be used in either two-dimensional orthree-dimensional simulations. Shell elements have both length and width and mayonly be used in three-dimensional simulations.

5.3 Governing Equations for Fluid Flow and Heat TransferProblems

The numerical modeling process begins with a physical model that is based ona number of simplifying physical assumptions. These assumptions can be madein light of the understanding of the physical processes that are involved. A math-ematical model is built from the physical model and then a series of PDEs aredeveloped.

For most of the practical physical processes, these PDEs cannot be solvedanalytically but must be solved numerically. Analytic solutions have been foundonly for a small subset of all possible fluid flow problems. Nonlinear nature ofthe governing equations can produce extremely complex flow fields and multipletime-dependent solutions, even for simplest geometries. Normally, the nature ofthe mathematical model would dictate the type of numerical model employed and areview of the current literature would suggest the most appropriate numerical modelto use. A combination of the mathematical model and numerical model leads toa set of governing equations. Before these governing equations can be solved,



a solution algorithm should be developed that can solve the equations correctly andefficiently.

In this section, the governing equations and auxiliary equations are focusedupon for the fluid flow and heat transfer phenomena and dynamic moving boundaryproblems. The details are discussed as following aspects:

(a) Formation of the set of nonlinear coupled PDEs for mass and momentumbalances and various types of boundary conditions.

(b) Handling method for four types of terms in the governing equations, includingtransient, advection, diffusion, and source.

(c) Uniform stabilized GLS formulation, including SUPG method and PSPGmethod, which demonstrates viable equal-order velocity and pressure inter-polations.

5.3.1 General form of governing equations

In this section, the strong form of the fluid flow problem is presented using a linearconstitutive relationship, leading to a set of nonlinear coupled PDEs for mass andmomentum balances. Various types of boundary conditions are also introduced.

Let and (0,T) denote the spatial and temporal domains, with x and t repre-senting the coordinates associated with and (0,T). The boundary of the domain may involve several internal boundaries.

Conservation of massIn solving fluid flow problems, it is important that the mass is conserved in thesystem. The continuity equation may be written as

∂ρ

∂t+ ∇ · (ρ V ) = 0 (5)

where the term ∂ρ/∂t stands for the change in density over time; it also describesthe shrinkage-induced flow in heat transfer solidification.

Conservation of momentumThe differential equation governing the conservation of momentum in a givendirection for a Newtonian fluid can be written as

∂(ρ V )

∂t+ ∇ · (ρ V V ) = ∇ · (µ∇ V ) − ∇P + B + Sm (6)

where µ is the dynamic viscosity, P is the pressure, B is the body force per unitvolume, and Sm is momentum source term.

The component form of equation (6) in 3D is

ρ

(∂u1

∂t+ u1

∂u1

∂x1+ u2

∂u1

∂x2+ u3

∂u1

∂x3

)−µ

(∂2u1

∂x21

+ ∂2u1

∂x22

+ ∂2u1

∂x23

)+ ∂p

∂x1−f1 = 0

(7)



ρ

(∂u2

∂t+ u1

∂u2

∂x1+ u2

∂u2

∂x2+ u3

∂u2

∂x3

)−µ

(∂2u2

∂x21

+ ∂2u2

∂x22

+ ∂2u2

∂x23

)+ ∂p

∂x2−f2 = 0

(8)

ρ

(∂u3

∂t+ u1

∂u3

∂x1+ u2

∂u3

∂x2+ u3

∂u3

∂x3

)−µ

(∂2u3

∂x21

+ ∂2u3

∂x22

+ ∂2u3

∂x23

)+ ∂p

∂x3−f3 = 0

(9)If the flow is turbulent, the time-averaged N–S equations have the same form

as equation (6), except that the fluid viscosity is replaced by an effective viscosity,µeff , which is defined as

µeff = µ+ µt (10)

Here µ is still the fluid dynamic viscosity while µt is the turbulence viscosity.

Conservation of energyThe equation governing the conservation of energy in terms of enthalpy can bewritten in general as

∂(ρH )∂t

+ ∇ · (ρ V H ) = ∇ · (k∇T ) + Sh (11)

where H is the specific enthalpy, k is the thermal conductivity, T is the temperature,and Sh is the volumetric rate of heat generation. The first term on the right-handside of equation (11) represents the influence of conduction heat transfer within thefluid, according to the Fourier’s law of conduction.

The energy equation could also be written in terms of static temperature; how-ever, the enthalpy form of energy equation generally considers the multiphaseflow such as steam/water, moist gas flow, and phase change like solidification andmelting.

Transport equationsThe general transport equation for advection–diffusion is described as

∂∂t (ρφ) + ∇(ρ V · φ) = ∇(ρ · ∇φ) + S (12)

Here, φ is the phase quantity in general expression, is the diffusion coefficientin general expression, the first left-hand term is the temporal term, and the secondleft-hand term is the convective term. The first right-hand term is the diffusive termand the second right-hand term is the source term.

Boundary conditionsDifferent types of boundary conditions can be encountered in fluid flow problems.The simplest one could be the Dirichlet boundary condition, which is given by

u = ug (13)



Another boundary condition is traction-free boundary condition, which is oftenapplied for the far downstream boundary of the flow domain and is given by

σ · n = 0 (14)

For some special cases, traction force can also be applied as boundaryconditions.

For solid boundaries or symmetric planes, the normal component of velocitymust be specified as “no penetration” condition, i.e.,

u · n = un = 0 (15)

The shear stress at the symmetric planes must be specified as a natural boundarycondition as follows:

µ∂uτ∂n

= 0 (16)

To be more specific, for a Newtonian fluid and a straight edge, equation (16)can be rewritten in terms of its normal and tangential components:

σn = (n · σ) · n = −p + 2µ∂un

∂n(17)

στ = (n · σ) · τ = µ(∂uτ∂n

+ ∂un

∂τ

)(18)

If the symmetry line is aligned with a Cartesian axis, equation (15) becomesa Dirichlet boundary condition while equation (16) is satisfied by setting στ = 0.Thus, a symmetry line involves both Dirichlet and Neumann type of boundaries.

5.3.2 Discretized equations and solution algorithm

This section is used to reduce the governing PDEs to a set of algebraic equations. Inthis method, the dependent variables are represented by polynomial shape functionsover a small area or volume (element in finite-element method). These represen-tations are substituted into the governing PDEs and then the weighted integral ofthese equations over the element is taken where the weight function is chosen tobe the same as the shape function. The result is a set of algebraic equations for thedependent variable at discrete points or nodes on every element.

General form of discretization equationsThe momentum, energy, temperature, and turbulent equations all have the form ofa scalar transport equation. There are four types of terms in the equation, transient,advection, diffusion, and source. For the purpose of describing the discretization



Table 5.2. General meaning of transportation representation

φ Meaning Cφ φ Sφ

u X velocity 1 µe ρgx − ∂p

∂x+ Rx

v X velocity 1 µe ρgx − ∂p

∂x+ Rx

w X velocity 1 µe ρgx − ∂p

∂x+ Rx

T Temperature Cp K Qv + Ek + W v + µ+ ∂p/∂tk Turbulence kinetic 1 µt/σk µt/µ− ρε+ C4βgi

(∂T

∂xi

)/σt

energy

ε Turbulence dissipation 1 µt/σe C1µtε/k − C2ρε2/k+

rateC1CµC3βkgi

(∂T

∂xi

)/σt

methods, the variable is referred to as φ. Then the form of the scalar transportequation is

∂

∂t(ρCφφ) + ∂

∂t(ρuCφφ) + ∂

∂t(ρvCφφ)

∂

∂t(ρwCφφ)

(19)

= ∂

∂x

(φ∂φ

∂x

)+ ∂

∂y

(φ∂φ

∂y

)+ ∂

∂z

(φ∂φ

∂z

)+ Sφ

whereCφ is the transient and advection coefficient, φ is the diffusion coefficient,and Sφ are source terms.

The pressure equation can be derived from continuity equation by segregatedsolution scheme, which is discussed in the next chapter. Equation (19) is a generalform of velocity, temperature, turbulent variables k , ε, and species. Table 5.2 liststhe corresponding equation representation for Cφ,φ, and Sφ.

The discretization process, therefore, consists of deriving the element matricesto put together the matrix equation in simplified matrix form as follows:

([ATransiente ] + [AAdvection

e ] + [ADiffusione ])φe = Sφe (20)

Galerkin method of weighted residuals is used to form the element integrals,denoted by N e the weighting function for the element, which is also the shapefunction.



Temporal termFor transient analyses, the transient terms are discretized using an implicit or back-ward difference method. Using the matrix algebra notation, a typical steady-statescalar transport equation with momentum, energy, and turbulence variables can bewritten:

Aijuj = Fi

where Aij contains the discretized advection and diffusion terms from the gov-erning equations, uj is the solution vector or values of the dependent variable(u, v, w, T , K , . . . .) and Fj contains the source terms.

The transient terms in the governing equations took the form ρ∂φ/∂t, whereφ represents the dependent variable (u, v, w, . . . .). This term is discretized using abackward difference method:

∂φ

∂t= φnew − φold

t(21)

Add this term to the matrix equation above

(Aij + Bij)unewj = Fi + Biiu

oldj (22)

where Bij is a diagonal matrix composed of terms like:

Bii = 1

t

∫Niρde (23)

This discretized transient equation must be solved iteratively at each time step todetermine all of the new variables (variable values at the latest time).

Advection termThere are two approaches to discretize this most challengeable term in fluid flow:the monotone streamline-upwind (MSU) approach is the first-order accurate andproduces smooth and monotone solutions, and the SUPG approach is the second-order accurate.

Monotone streamline upwind (MSU) approach. In the MSU method, the advec-tion term is handled through a monotone streamline approach based on the idea thatadvection transport is along characteristic lines. It is useful to think of the advectiontransport formulation in terms of a quantity being transported in a known velocityfield. The velocity field itself could be envisioned as a set of streamlines everywheretangent to the velocity vectors. The advection terms can be expressed in terms ofthe streamline velocities.

In advection-dominated transport phenomena, one assumes that no trans-port occurs across characteristic lines, i.e., all transfer occurs along streamlines.



Therefore, if one assumes that the advection term is expressed in streamlinedirection:

∂(ρcφvxφ)

∂x+ ∂(ρcφvyφ)

∂y+ ∂(ρcφvzφ)

∂z= ∂(ρcφvsφ)

∂s(24)

then the advection term is constant throughout an element:

[Aadvectione ] = ∂(ρcφvsφ)

∂s

∫Nde (25)

Figure 5.2 illustrates the streamline across a brick element; this formulationcould be used for every element, each of which will have one node which getscontributions from inside the element; the derivative is calculated using a simpledifference as follows:

∂(ρcφvsφ)

∂s= (ρcφvsφ)up − (ρcφvsφ)down

s(26)

wheredown = Subscript for value at downstream nodeup = Subscript for value taken at the location at which the streamline through thestreamline through the upwind node enters the element. The value is interpolatedin terms of the nodes in between.s = Distance from the upstream point to the downstream node.

The whole process goes through all the elements and identifies the downwindnodes. The calculation is made based on the velocities to find where the streamlinethrough the downwind node came from. Weight factors are calculated based on theproximity of the upwind location to the neighboring nodes.

Streamline-upwind/Petrov–Galerkin (SUPG) approach. This method is usedto discretize the advection term and an additional diffusion-like perturbation term,which acts only in the advection direction.

For three-dimensional problem, the perturbed interpolation function is given by

Wi = Ni +(βh

2| V |

)(u∂Ni

∂x+ v∂Ni

∂y+ w

∂Ni

∂z

)(27)

Here, V = (u, v, w) is the velocity field in three-dimensional domain, h is theaveraged size of an element, which, for eight-node brick element shown in Figure5.2, can be calculated as

h = 1

| V | (|h1| + |h2| + |h3|) (28)



1

2

8

7

6

5

4

3

Streamline

fdown

x

zy

fup

DS

Figure 5.2. Streamline upwind approach for 3D brick element.

where

h1 = a · V , h2 = b · V , h3 = c · V (29)

Taking an example of a eight-node brick element, the element direction vectors,a = (ax, ay, az), b = (bx, by, bz), and c = (cx, cy, cz) are calculated using the brickelement nodal coordinates as shown in Figure 5.2.

ax = 1

2(x2 + x3 + x6 + x7 − x1 − x4 − x5 − x8)

ay = 1

2(y2 + y3 + y6 + y7 − y1 − y4 − y5 − y8)

az = 12

(z2 + z3 + z6 + z7 − z1 − z4 − z5 − z8)

bx = 1

2(x5 + x6 + x7 + x8 − x1 − x2 − x3 − x4)

by = 1

2(y5 + y6 + y7 + y8 − y1 − y2 − y3 − y4)

bz = 1

2(z5 + z6 + z7 + z8 − z1 − z2 − z3 − z4)

cx = 1

2(x1 + x2 + x5 + x6 − x3 − x4 − x7 − x8)

cy = 1

2(y1 + y2 + y5 + y6 − y3 − y4 − y7 − y8)

cz = 1

2(z1 + z2 + z5 + z6 − z3 − z4 − z7 − z8)



Diffusion termThe diffusion term is obtained by integration over the problem domain after it ismultiplied by the weighting function.

[Adiffusione ] =

∫N∂

∂x

(φ∂φ

∂x

)de+

∫N∂

∂y

(φ∂φ

∂y

)de +

∫N∂

∂z

(φ∂φ

∂z

)de

(30)

Once the derivative of φ is replaced by the nodal values and the derivativesof the weighting function, the nodal values will be removed from the integrals,∂φ∂x = ∂Ni

∂x φ, for momentum equations, the diffusion terms in x, y, z direction aretreated in similar fashion, a typical diffusion item in momentum equation will beKij =

∫eµ∂Nm∂xj

∂Nn∂xi

de.

Source termSource term consists of merely multiplying the source terms by the weightingfunction and integrating over the volume as follows:

Seφ =

∫N iSφde (31)

5.3.3 Stabilized method

The difficulties mostly encountered in finite-element method are from the mesh thatis generated by general automatic mesh engine. Given a three-dimensional compli-cated geometry, which is usually from CAD package, e.g., the amount of elementsfor automatic generated purely brick could be huge and makes the model analysisimpractical. So hybrid meshes are practically used, but this approach involves lotof nonbrick elements such as tetrahedron, wedge, and pyramid. This also happensin two-dimensional hybrid element with quadrilateral and triangle elements.

However, since all the two-dimensional and three-dimensional degenerated ele-ments do not satisfy the LBB condition, the fluid flow velocity and pressure solutiongets locked. The LBB condition restricts the type of element that can be used forthe Galerkin FEM formulation. Both two-dimensional linear triangular element andthree-dimensional linear tetrahedral, wedge, and pyramid element do not satisfy thiscondition.

Approaches to handle LBB conditionThere are two approaches to overcome the above difficulties: one is to use high-orderelements and the other to use stabilized FEM based on mixed formulation.

Approach one: high-order elements. High-order elements use midside nodes toperform quadratic interpolation over elements, but the big disadvantage is that thetotal unknowns will increase greatly due to the extra mid-nodes at the elementedges, making the model size limitation to be more restricted – for this reason, thehigh-order element is rarely used in commercial finite-element codes.



Approach two: stabilized mixed FEM formulation. Although the mixed for-mulation requires the elements to be LBB type, there are lot of research work andefforts to develop new techniques that circumvent this restriction. These techniquesinclude bubble function method, two-level mesh method, and the GLS formulations.These techniques allow the usage of those types of elements that do not satisfy theLBB condition, such as two-dimensional triangular element (linear velocity/linearpressure) and three-dimensional tetrahedron, wedge, and pyramid element (lin-ear velocity/linear pressure interpolation); this is also referred to as equal-orderformulation.

Step 1: Taking three-dimensional incompressible flow as an example for themixed FEM–based fluid flow formulation:⎧⎪⎪⎨⎪⎪⎩

F1F2F30

⎫⎪⎪⎬⎪⎪⎭ =

⎡⎢⎢⎣M 0 0 00 M 0 00 0 M 00 0 0 0

⎤⎥⎥⎦⎧⎪⎪⎨⎪⎪⎩

u1u2u3

P

⎫⎪⎪⎬⎪⎪⎭ +

⎡⎢⎢⎣C(u) 0 0 0

0 C(u) 0 00 0 C(u) 00 0 0 0

⎤⎥⎥⎦⎧⎪⎪⎨⎪⎪⎩

u1u2u3P

⎫⎪⎪⎬⎪⎪⎭

+

⎡⎢⎢⎣2K11 + K22 + K33 K12 K13 −Q1

K21 K11 + 2K22 + K33 K23 −Q2K31 K32 K11 + K22 + 2K33 −Q3

−QT1 −QT

2 −QT3 0

⎤⎥⎥⎦⎧⎪⎪⎨⎪⎪⎩

u1u2u3P

⎫⎪⎪⎬⎪⎪⎭(32)

A more compact form of this equation is given as

[M 00 0

]uP

+

[C(u) + K(u) QT

−QT 0

]uP

=

F0

(33)

or in a more symbolic format as

MU + KU = F (34)

The nature of the matrix is sparse, generally unsymmetric, and positive semidefi-nite. The zero entries on the diagonal of the matrix lead to difficulties for the solutionof the system of algebraic equations. Direct solver needs to have pivoting technique.

Step 2: Implementation of equal-order elements for the mixed FEM–based codefirst, in terms of the implementation of so-called bubble function to stabilize thescheme.

In this approach, the basic formulation is still the Galerkin FEM, but velocityinterpolation is carried out with one additional DOF, which is the centroid of tri-angular element (2D) and tetrahedral element (3D). The bubble function for thecentral node velocity can be chosen as a piecewise linear, with its value being zeroat the element boundaries and piecewise linear on two-dimensional subtriangles



Figure 5.3. Bubble function for linear triangular and tetrahedral elements.

or three-dimensional subtetrahedron. The pressure is interpolated linearly over theelement (Figure 5.3).

Step 3: Implementation of stabilized FEM formulation GLS for two-dimensional and three-dimensional fluid flow using two-dimensional and three-dimensional degenerated elements, i.e., using equal-order interpolation (linearvelocity/linear pressure) for both velocity and pressure.

The GLS variation form of the momentum and continuity equation is given by∫

w · (ρ0u · ∇u)dx +∫

∇w · (−P + 2µD(u))dx −∫

ρ0wfdx + Q∇ · udx

+nel∑

n=1

∫n

RGLSdx = −∫

wTids (35)

Here, w and Q are the weight functions for the momentum equation and thecontinuity equation, respectively. The element residual is defined as

RGLS = (τSUPGρ0u · ∇w + τPSPG∇Q − τGLS2µD(w))

· (ρ0u · ∇u + ∇P − 2µD(u)) (36)

The τ coefficients are weighting parameters, which could be functions of theelement size, element Reynolds number, and/or element average velocity. If thesecoefficients are selected properly, equal-order elements can be used and LBBcondition is not necessary.

For momentum equation, the final equations are given by

(C(u) + CSUPG(u) + CGLS(u))u + (K + KSUPG + KGLS)u

− (Q + QSUPG + QGLS)P = F (37)

For continuity equation, we have

−QTu − QPSPGP + CPSPG(u)u + KPSPGu = 0 (38)



Galerkin/least-squares formulationThe stability of the mixed Galerkin finite element for incompressible flows can beenhanced by the addition of various least-squares terms to the original Galerkinvariational statement. The Galerkin/Least-squares approach is sometimes termeda residual method because the added least-squares terms are weighted residualsof the momentum equation; this form of the least-squares term implies the con-sistency of the method since the momentum residual is employed. The GLS – aPetrov–Galerkin method – is also known as a perturbation method since the addedterms can be viewed as perturbations to the weighting functions. Development andpopularization of the GLS methods for flow problems are primarily due to Hughes,Tezduyar, and co-workers and are a generalization of their work on SUPG andPSPG methods.

The usual approach to the GLS begins with the discontinuous approximation intime Galerkin method and considers a finite-element approximation for a space–time slab. Continuous polynomial interpolation is used for the spatial variationwhile a discontinuous time function is used within the space–time slab.The assumedtime representation obviates the need to independently consider time integrationmethods. To simplify the present discussion of the GLS, the time independent formof the incompressible flow problem is considered which avoids the complexity ofthe space–time finite-element formulation. Using vector notation and followingthe weak form development, the GLS variational form for the momentum andcontinuity equations can be written as∫

Nm(ρu · ∇u)dx +∫

∇Nm · ( − P + 2µ∇D(u)dx −∫

ρNmfxdx +∫

Nn∇ · u)dx

+nel∑

n=1

∫

RGLS = −∫

Nmτidsdx (39)

where Nm and Nn are the weight functions for the momentum and continuityequations and the element residual is defined by

RGLS = (δ+ ε+ β) · (ρu · ∇u + ∇(P − 2µ∇D(u)) (40)

δ= τSUPGu · ∇Nm

ε= τPSPG1ρ

∇Nn

β= − τGLS2µ∇(D(Nm))The τ coefficients are weighting parameters; if the definitions for δ, ε, and β

are substituted into equation (40) and the various weighting parameters are madeequivalent to a single parameter τ, the RGLS will be

RGLS = τ[ρu · ∇Nm + ∇Nn − 2µ∇ · D(u)] · [ρu · ∇u + ∇(P − 2µ∇ · D(u)] (41)



This is the standard definition for the residual contribution to the GLS formula-tion. The splitting of the first residual into three separate contributions in equation(41) was done to allow the original SUPG and PSPG formulations to be easilyrecovered.

If β and ε are set to zero, the stabilized SUPG method is recovered.If β and δ are set to zero, the stabilized PSPG method is recovered.If only β is set to zero, both SUPG and PSPG stabilization methods are

recovered.Note that in equation (39) the first four integrals and the right-hand side define a

standard Galerkin-weighted residual method that is written in terms of global shapefunctions; the integrals are over the simulation domain which is composed by thediscretized finite elements. The added residual term in equation (39) is defined overthe interior of each element and basically contains the square of all or parts of themomentum residual. The various τ parameters are positive coefficients that havethe dimension of time. The forms of these parameters are usually developed fromerror estimates, convergence proofs, and dimensional analysis. Particular constantswithin each parameter are selected by optimizing the method on simple problemsand generalizing to multidimensions. It should be noted that the development ofthe τ parameters is not a unique process. For the steady problems considered herea typical τ for the GLS method is

τ =[(

2‖u‖h

)2

+(

4ν

h2

)2]−1/2

(42)

where h is an appropriate element length. For the SUPG and PSPG formulations,the τ values are the function of an element Reynolds number and the ratio of anelement length to a velocity scale.

The error and convergence analyses for the GLS and PSPG methods havedemonstrated that the equal-order velocity and pressure interpolation for thesemethods is viable and the LBB condition is not required [6]. In finite-elementimplementation, convenient elements, such as the bilinear Q1/Q1, linear P1/P1,biquadratic Q2/Q2, and quadratic P2/P2, are now useful approximations whilebeing unstable in the Galerkin finite-element context. The low-order elements havean advantage over their higher-order counterparts because some of the stabiliza-tion terms associated with viscous diffusion are identically zero. This considerablysimplifies the element equation building process.

The matrix form of velocity–pressure equationsThe matrix form of the stabilized GLS formulation could be summarized as theMomentum equation and Continuity equation, respectively.Momentum: (C(u) + Cδ(u) + Cβ(u)u + (K + Kδ+ Kβ)u − (Q + Qδ+ Qβ)P = FContinuity: − QT u − QεP + (Cε(u) + Kε)u = 0

Expended discretization form of equations for velocity–pressure coupling andSUPG stabilization, which are summarized in Table 5.3, will be used as the base



Table 5.3. Element-level matrices

M =∫e

ρNmNnde C(u) =∫e

ρNmuj∂Nn

∂xjde

Su(u) =∫e

[ρτsui

∂Nm

∂xi

]uj∂Nn

∂xjde Spi(u) =

∫e

[τsuj

∂Nm

∂xj

]∂Nn

∂xide

Kij =∫e

µ∂Nm

∂xj

∂Nn

∂xide Qi =

∫e

∂Nm

∂xiNnde

Sδij =∫e

ρδ∂Nm

∂xi

∂Nn

∂xjde Kτi (u) =

∫e

τp∂Nm

∂xi

[uj∂Nn

∂xj

]de

Qτ =∫e

τp

ρ

[∂Nm

∂xj

∂Nn

∂xj

]de SFi(u) =

∫e

ρτs

[uj∂Nm

∂xj

]de fbi

SF4 =∫e

τp

[fbj∂Nm

∂xj

]de Fi =

∫∂e

Nmide +∫∂e

ρNm fBide

for our further segregated formulation in the next chapter.

⎡⎢⎢⎣M1

M2M3

0

⎤⎥⎥⎦⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

˙u˙v˙w˙p

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

+

⎡⎢⎢⎢⎣C(u) + Su(u) Sp1(u)

C(u) + Su(u) Sp2(u)C(u) + Su(u) Sp3(u)

−Kτ1 (u) −Kτ2 (u) −Kτ3 (u) 0

⎤⎥⎥⎥⎦⎧⎪⎪⎪⎨⎪⎪⎪⎩

uvwp

⎫⎪⎪⎪⎬⎪⎪⎪⎭

+

⎡⎢⎢⎢⎣2K11 + K22 + K33 K12 K13 −Q1

K21 K11 + 2K22 + K33 K23 −Q2

K31 K32 K11 + K22 + 2K33 −Q3

−QT1 −QT

2 −QT3 0

⎤⎥⎥⎥⎦⎧⎪⎪⎪⎨⎪⎪⎪⎩

uvwp

⎫⎪⎪⎪⎬⎪⎪⎪⎭

+

⎡⎢⎢⎢⎣Sδ11 Sδ12 Sδ13 0Sδ21 Sδ22 Sδ23 0Sδ31 Sδ32 Sδ33 0

0 0 0 −Qτ

⎤⎥⎥⎥⎦⎧⎪⎪⎪⎨⎪⎪⎪⎩

uvwp

⎫⎪⎪⎪⎬⎪⎪⎪⎭ =

⎧⎪⎪⎪⎨⎪⎪⎪⎩F1

F2

F3

0

⎫⎪⎪⎪⎬⎪⎪⎪⎭ +

⎧⎪⎪⎪⎨⎪⎪⎪⎩SF1 (u)SF2 (u)SF3 (u)−SF4

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (43)

It is noted that since all of the SUPG τ parameters depend on the elementsize, in the limit as the element size gets smaller, the stabilized method reduces to



the Galerkin form. In another words, stabilization plays a significant role only forcoarse mesh.

5.4 Formulation of Stabilized Equal-OrderSegregated Scheme

This section discusses the strategies for the solution of large systems of equationsarising from the finite-element discretization of the formulations. To solve thenonlinear fluid flow and heat transfer problem, particular emphasis is placed onsegregated scheme in nonlinear level and iterative methods in linear level. Thereality of velocity–pressure coupling formulation in the last section not only resultsin a large dimensioned system but also generates a stiffness matrix radically differentto the narrowband type of matrix, which is inefficient to solve and the advectionterm causes solution instability.

The first part in the section is the kernel part; it is a uniformed consistent solutionscheme for nonlinear and linear level to support any fluid flow/heat transfer andtransport phenomena. The combination of “segregated scheme,” “iterative solver,”and “Block I/O” composes the whole solution picture in a very efficient and robustway. First, the decoupled pressure–velocity segregated scheme is discussed andexpended in very detail under finite-element method by nonlinear level, and thenthe Generalized Minimum Residual Method (GMRES)-based iterative techniqueand matrix preconditioners are used to solve linear equations. With respect to imple-mentation, the condensed storage and block I/O technique is demonstrated to boostsolution efficiency especially for large systems.

5.4.1 Introduction

Several categorizations can be made by using finite-element method to solve theNavier–Stokes equation for fluid flow and thermal problems. They are mainly(a) velocity–pressure coupling and (b) orders of interpolation functions forvelocity and pressure.

The first categorization is based on how the primitive variables of velocityand pressure are treated. The solution schemes are categorized into three groups:velocity–pressure integrated method, penalty method, and segregated velocity–pressure method. The velocity–pressure integrated scheme treats velocity andpressure simultaneously; it needs less iteration number in nonlinear level, but largerglobal matrix requires longer execution time and a large memory in linear level; andusually its overall runtime performance is slow. The penalty scheme eliminates thedirect pressure computation by introducing penalty function, thus speeding up theexecution time and memory requirement; however, pressure field from simple post-process is not satisfying especially for large Reynolds number fluid flow problem.Recently, much attention has been paid to the segregated velocity–pressure scheme,where velocity and corresponding pressure field are computed alternatively in aniterative sequence. This scheme has the advantage of speed and memory usage with



the price of complicated formulation and possible less solution robustness due tovelocity–pressure decoupling (addressed later).

The second categorization is based on the mixed-order interpolation and equal-order interpolation according to the orders of interpolation functions for the velocityand pressure in element level. The mixed-order interpolation attempts to eliminatethe tendency to produce checkerboard pressure pattern and to satisfy LBB condi-tion; the velocity in this method is interpolated linearly, whereas the pressure isassumed to be constant; this interpolation is referred to as Q1P0. The equal-orderinterpolation in Ref. [5] uses linear interpolation for velocity–pressure formulation(referred to as Q1P1) without exhibiting spurious pressure modes, by employingnonconsistent pressure equations (NCPEs) for pressure correction. This methodmakes the solver more effective than constant element pressure in an element forQ1P0 method.

The objective of this chapter is to explore further the merits of the finite-elementmethods in simulating the flow where the upstream effects play important roles.Thus, we are presenting the details of making use of both the segregated velocity–pressure and equal-order formulations on the basis of SUPG method.

The basic idea of segregated algorithms is to decouple the pressure calculationfrom the velocity calculation by taking the divergence of the vector momentumequation and applying some clever insights regarding incompressible flow. Earlymotivation for this approach was largely twofold: to mitigate memory requirementsof fully coupled algorithms and to enable semi-implicit time integration.

Pressure–velocity segregation methods have been reviewed by severalresearchers in the context of the finite-element method; most notable are the papersby Gresho and Haroutunian et al. [2]. Basically, all current segregated algorithmvariants are distinguished by the way in which the pressure is decoupled and pro-jected from one time step to the next. Haroutunian and Engelman [2] proposed threeconsistent finite-element counterparts to the SIMPLE and SIMPLER algorithm.To further reduce the size of the submatrix systems, each individual componentof the momentum equations was solved separately and successively by iterativetechniques. Overall at each Newton iteration or Picard iteration they solved fourmatrix subsystems, one for each of three velocity components and one for thepressure. Interestingly, the most challenging matrix system to solve happens to beone arising from the discretization of the pressure equation; here, the right-handside SF4 in equation (5.43) is lagged from the last iteration so that this equationis solved solely for the pressure. The resulting matrix, despite being symmet-ric, is actually very poorly conditioned due to poor scaling. Nonetheless, thesechallenging matrix systems can be readily solved by modern iterative solvers andreordering/preconditioner strategies, and the success of this algorithm over the lastdecade has been enormous.

In our view this approach is still a compromise to the favorable convergenceproperties of a fully coupled technique advocated here. Convergence to stable solu-tions at successive time steps is linear at best case and sometimes asymptotic,sometimes resulting in large number of required segregation iterations (albeit fastiterations). Moreover, the method introduces several relaxation parameters that



must be tuned to the application. If one chooses alternatives to the primitive vari-able formulation, then boundary conditions on velocity become challengeable toapply accurately. Finally, codes centered around this algorithm are more complex instructure, as they still contain the intricacies of matrix solution services but involvemore than one data structure (for the mixed interpolation) and more inner and outeriteration loops.

5.4.2 FEM-based segregated formulation

This formulation is based on the algorithm of Rice and Schnipke’s [5] and Shaw’s [7]pioneer works; they use non-consistent pressure equations (NCPE) for the pressurestep and pressure correction step rather than the conventional weak form of the flowequations resulting from Galerkin finite element context. This approach has gainedpopularity in recent years with the very desirable advantage of precluding spuriouspressure modes from pressure solution. This consequently allows them to employthe desirable feature of “equal-order interpolation for the velocity and pressure.”For this reason, the algorithm can solve the flow without using very refined meshesand hybrid element types highly desirable for commercial code obtained from CADmodel. However, the expanse of this algorithm includes the extra work of imposingboundary conditions properly and their complex formulation. In this section, weshow the formulation derivation details starting from the N–S equation.

Bilinear interpolationThe fluid flow continuity equation and the momentum equation follow same formas listed in Section 3.1. For implementation using finite-element method, velocityand pressure at arbitrary position could be interpolated using “equal-order bilinearinterpolation” to adapt brick or other degenerated elements.

u(x, y) = Niui (44a)

v(x, y) = Nivi (44b)

w(x, y) = Niwi (44c)

p(x, y) = Nipi (44d)

Discretization and system assemblyUsing the standard Galerkin approach, the momentum equation is multiplied byweighting functions, and using the Green–Gauss theorem, the diffusion terms areintegrated by parts in x momentum equation as follows:∫

ρN

(u∂u

∂x+ v∂u

∂y+ w

∂u

∂z

)d = −

∫

N∂p

∂xd+

∫

NFxd

(45)

+ µ∫

N

[∂2u

∂x2 + ∂2u

∂y2 + ∂2u

∂z2

]d



Construct a function,

∫

⎡⎢⎢⎣∂(∂u

∂xN

)∂x

+∂

(∂u

∂yN

)∂y

+∂

(∂u

∂zN

)∂z

⎤⎥⎥⎦d(46)

=∫

(∂u

∂x

∂N

∂x+ ∂u

∂y

∂N

∂y+ ∂u

∂z

∂N

∂z

)d+

∫

N

(∂2u

∂x2 + ∂2u

∂y2 + ∂2u

∂z2

)d

Using Green–Gauss theorem,

∫

⎡⎢⎢⎣∂(∂u

∂xN

)∂x

+∂

(∂u

∂yN

)∂y

+∂

(∂u

∂zN

)∂z

⎤⎥⎥⎦d =∫

N

(∂u

∂xnx + ∂u

∂yny + ∂u

∂znz

)d

(47)

Equations (45), (46), and (47) are combined to get the new form of thediscretized x-momentum equation:∫

ρN

(u∂u

∂x+ v∂u

∂y+ w

∂u

∂z

)d = −

∫

N∂p

∂xd+

∫

NFxd

+∫

N

(∂u

∂xnx + ∂u

∂yny + ∂u

∂znz

)d−

∫

(∂u

∂x

∂N

∂x+ ∂u

∂y

∂N

∂y+ ∂u

∂z

∂N

∂z

)d

Move the last item in the right-hand side to the left-hand side, then

LHS = [Au]u =∫

[ρN

(u∂u

∂x+ v∂u

∂y+ w

∂u

∂z

)+

(µ∂u

∂x

∂N

∂x+ µ∂u

∂y

∂N

∂y+ µ∂u

∂z

∂N

∂z

)]d

= ρ∫

Ni

[u∂Nj

∂x+ v∂Nj

∂y+ w

∂Nj

∂z

]du +

[∫

µ∂Ni

∂x

∂Nj

∂xd

]u

+[∫

µ∂Ni

∂y

∂Nj

∂yd

]u +

[∫

µ∂Ni

∂z

∂Nj

∂zd

]u

= [C(u, v, w) + K11 + K22 + K33]u (48a)

where, C(U ) is the advection item and K11, K22, K33 are the diffusion items.To obtain C(U ), velocities u, v, w are considered the average values in an element.

RHS = −∫

Ni∂p

∂xd+

∫

NiFxd+∫

Ni

(∂u

∂xnx + ∂u

∂yny + ∂u

∂znz

)d = −M14p + FxA + Fx (48b)

[Av] and [Aw] could be achieved by similar approaches.



SUPG items The SUPG weighting function proposed by Brooks is defined by:Wk = Nk + pk

Here the streamline-upwind contribution pk is defined as:

pk = ke

(ue1)2 + (ue

2)2

(ue

j∂Ni

∂xj

)(49)

Only convective items are weighted with the SUPG functions for the inconsistentapproach, which is believed to be more stable than the consistent SUPG formulation[10]. The extra items from pk should be added into K11, K22, K33 in equation (49)in the implementation.

Equation of pseudo velocity (prediction phase)In matrix form of the segregated momentum equations:

[M11]u = −[M14]p + FxA + Fx (50a)

[M22]v = −[M24]p + FyA + Fy (50b)

[M33]w = −[M34]p + FzA + Fz (50c)

where

[M11] = [C(u, v, w) + K11 + K22 + K33] + [SUPG(u, v, w)]



[M14] =∫

Ni∂p

∂xd

[M24] =∫

Ni∂p

∂yd

[M34] =∫

Ni∂p

∂zd

FxA =∫

Ni

(∂u

∂xnx + ∂u

∂yny + ∂u

∂znz

)d

FyA =∫

Ni

(∂u

∂xnx + ∂u

∂yny + ∂u

∂znz

)d

FzA =∫

Ni

(∂u

∂xnx + ∂u

∂yny + ∂u

∂znz

)d

Fx =∫

NiFxd

Fy =∫

NiFyd

Fz =∫

NiFzd

where is the fluid domain (volume); and is the flow boundaries (surface).There are no SUPG items in FxFyFz for the inconsistent approach.



The above three equations can be solved separately to obtain the velocityestimate (prediction phase).

Hat Velocity (Vhat) Rewrite equations (50a), (50b), (50c):

auiiui = −∑i =j

auijuj −∫

Ni∂p

∂xd+ µ

∫

Ni

[∂u

∂xnx + ∂u

∂yny + ∂u

∂znz

]d+ Fxi

(51a)

aviivi = −∑i =j

avijvj −∫

Ni∂p

∂yd+ µ

∫

Ni

[∂v

∂xnx + ∂v

∂yny + ∂v

∂znz

]d+ Fyi

(51b)

awiiwi = −∑i =j

awijuj −∫

Ni∂p

∂zd+ µ

∫

Ni

[∂w

∂xnx + ∂w

∂yny + ∂w

∂znz

]d+ Fzi

(51c)

For constant pressure gradient in an element, rewrite equation (51a):

ui = 1auii

⎡⎣−∑i =j

auijuj + µ∫

Ni

[∂u

∂xnx + ∂u

∂yny + ∂u

∂znz

]d+ Fxi

⎤⎦− 1

auii

[∫

Nid

]∂p

∂x

i.e.,

ui = ui − kpi∂p

∂x(52a)

where

kpui = 1

auii

∫

Nid (53a)

ui = 1

auii

⎡⎣−∑i =j

auijuj + µ∫

Ni

[∂u

∂xnx + ∂u

∂yny + ∂u

∂znz

]d+ Fxi

⎤⎦ (54a)

Rewrite equation (51b):

vi = 1

avii

⎡⎣−∑i =j

avijvj + µ∫

Ni

[∂v

∂xnx + ∂v

∂yny + ∂v

∂znz

]d+ Fyi

⎤⎦− 1

avii

[∫

Nid

]∂p

∂y(50)

i.e.,

vi = vi − kpi∂p

∂y(52b)



where

kpvi = 1

avii

∫

Nid (53b)

vi = 1

avii

⎡⎣−∑i =j

avijvj + µ∫

Ni

[∂v

∂xnx + ∂v

∂yny + ∂v

∂znz

]d+ Fyi

⎤⎦ (54b)

Rewrite equation (51c):

wi = 1

awii

⎡⎣−∑i =j

awijwj + µ∫

Ni

[∂w

∂xnx + ∂w

∂yny + ∂w

∂znz

]d+ Fzi

⎤⎦− 1

awii

[∫

Nid

]∂p

∂z

i.e.,

w = wi − kpi∂p

∂z(52c)

where

kpwi = 1

awii

∫

Nid (53c)

wi = 1awii

⎡⎣−∑i =j

awijwj + µ∫

Ni

[∂w

∂xnx + ∂w

∂yny + ∂w

∂znz

]d+ Fzi

⎤⎦(54c)

To ensure mass continuity, the velocity should be updated (corrected) at eachiteration by which velocity correction is reformulated from the new pressure value,refer to equation 62a, 62b and 62c for velocity correction details.

VCorrected = Vhat + VCorrection_Term_By_Pressure (55)

Pressure Poisson equationOn applying Green–Gauss theorem and integration to the continuity equation,the integral of the divergence in a domain equals the net flux across the domainboundary: ∫

[∂(uN )∂x

+ ∂(vN )

∂y+ ∂(wN )

∂z

]d =

∫

N v · nd (56)

The continuity equation becomes:∫

N

(∂u

∂x+ ∂v

∂y+ ∂w

∂z

)d = −

∫

(∂N

∂xu + ∂N

∂yv+ ∂N

∂zw

)d

(57)+

∫

N (unx + vny + wnz)d = 0



The modified continuity equation (57) using green function transformation istargeted to solve pressure.

Step 1: Substitute interpolated velocity from equations (44a), (44b), and (44c)to continuity equation (57)

−∫

(∂Ni

∂x(Njuj) + ∂Ni

∂y(Njvj) + ∂Ni

∂z(Njwj)

)d+

∫

Ni(unx + vny + wnz)d = 0

(58)

Step 2: Substitute velocity prediction from equations (53a), (53b), and (53c)into equation (58)∫

[∂Ni

∂x

(NjK

pj∂p

∂x

)+ ∂Ni

∂y

(NjK

pj∂p

∂y

)+ ∂Ni

∂z

(NjK

pj∂p

∂z

)]d

=∫

(∂Ni

∂x(Njuj) + ∂Ni

∂y(Nj vj) + ∂Ni

∂z(Njwj)

)d−

∫

Ni(unx + vny + wnz)d

(59)

Step 3: Rewrite to final formNotice that p = Nipi, so substituting ∂p

∂x = ∂(Nipi)∂x = pi

∂Ni∂x (assume that p is con-

stant in an element here) in equation (59) (using similar way for and ) we obtain∂p∂y and ∂p

∂z .∫

[∂Ni

∂x(NkKp

k )(∂Nj

∂x

)]d+

∫

[∂Ni

∂y(NkKp

k )∂Nj

∂y

]d

+∫

[∂Ni

∂z(NkKp

k )∂Nj

∂z

]d

· p

(60)

=∫

∂Ni

∂x(Njuj)d+

∫

∂Ni

∂y(Nj vj)d+

∫

∂Ni

∂z(Njwj)d

−∫


Moving Vhat out of the integration in equation (60), we obtain:∫

[∂Ni

∂x(NkKp

k )(∂Nj

∂x

)]d+

∫

[∂Ni

∂y(NkKp

k )∂Nj

∂y

]d

+∫

[∂Ni

∂z(NkKp

k )∂Nj

∂z)]

d

· p

(61)

=∫

∂Ni

∂xNjdu +

∫

∂Ni

∂yNjdv +

∫

∂Ni

∂zNjdw

−∫


Now equation (61) is the final form of pressure Poisson equation.



It is interesting to note that the element pressure matrices are identical tothose obtained in classical diffusion-type problems, with the term K replacingthe diffusion coefficient.

Here are more discussions on segregated solution scheme.

(1) The coefficient matrix for the pressure equation is similar to that obtained forthe diffusion term in the conventional finite-element formulation if the viscosityis replaced by NiK

pj the formulation developed by Rice [5] and Vellando [12];

this fact is indicative of the stability and robust nature of the resulting pressurefield.

(2) The LHS and RHS of equation (69) could be constructed and evaluated inelement level and then assembled into global matrix LHS and RHS; this istypical finite-element behavior.

(3) In equation (60), kpi is a node-based property, it should be assembled from

the saved element-level matrix,∫

Nid, auii is also a global property fromassembly.

(4) Equation (60) is in the form of Poisson equation; the LHS matrix has theproperties of symmetric and positive definite as indicated by Shaw [7].

(5) Special boundary integrations on all inflow and outflow boundaries are requiredto solve pressure, which corresponds to the fourth item in equation (61).

Velocity correctionWrite equations (51a), (51b), and (51c) using weighting function p(x,y) = Nipi andchain law, and discarding velocity gradient items on boundaries. ui, vi, wi are usedfor the velocity prediction as “hat” velocity.

ui = 1

auii

⎡⎣−∑i =j

auijuj + Fxi

⎤⎦ − 1auii

[∫

Ni∂Nj

∂xd

]p

Or, ui = ui − 1

auii

[∫

Ni∂Nj

∂xd

]p = ui − 1

auiiM14p (62a)

Get corrected velocity v, w in the same way as u,

vi = vi − 1

avii

[∫

Ni∂Nj

∂yd

]p = vi − 1

aviiM24p (62b)

wi = wi − 1awii

[∫

Ni∂Nj

∂zd

]p = wi − 1

awiiM34p (62c)

At this step, u, v, w are directly corrected from the predicted and with trivialeffort, the second item is the pressure coupling effect on the velocity.

Special treatments on boundary conditionsThere are some special treatments in the segregated formulation.

Wall boundary: For wall boundary, the surface integral or natural boundarycondition for the pressure equation is zero. To incorporate the known velocities into



pressure equation, it is required that “hat” velocity be zero. In implementation, thepressure coefficients, kp

ui, kpvi, kp

wi at the wall nodes should be set as zero.Specified velocity boundary: The surface integration of velocity boundary

conditions should be added into the pressure equation (61)’s right hand side as in thefourth item in the right hand side; for specified velocity boundary, the contributionis fixed.

Inflow/Outflow boundary Zero gradient velocity boundary condition is appliedat the inflow/outflow boundaries (including inlet/outlet boundary and pressureboundary), i.e., no special treatment in the momentum equation is required atthis kind of boundary. However, the contribution from the surface integration ofnatural boundary conditions should be added to the right-hand side of the pressureequation as done in the fourth item on the right-hand side of equation (60).

Segregated scheme not only simply divides the big coupling matrix into smallermatrices, but also sets up a pressure equation (Poisson equation) with positive-definite symmetric properties, which can greatly expedite the pressure solutionwith better accuracy. Most of the fluid flow problems are driven by pressure, andtherefore the velocity components will follow a good solution of pressure in practice.

Solution stability controlInertial relaxation. Inertial relaxation is used in the governing equations to slowdown the convergence rate in the same manner as transit terms are used in nonsteadyproblems, so the inertial relaxation factor is also called virtual time step.

The inertial relaxation is added to the governing equations in the followingmanner: [

Aii + ρi∫

Nd

tinertia

]φi +

∑j =i

Aijφj = Fi + ρi∫

Nd

tinertiaφiold (63)

The second term in parentheses and the last term on the right-hand side are theinertial relaxation terms. tinertia can be automatically adjusted according to theglobal matrix condition number.

• Small tinertia value will enhance the solution stability and slow down theconvergence rate.

• tinertia is applied to momentum equations only, and it will indirectly affect thepressure equation because of the slow marching of the velocity change.

• For convergence solution, the inertial term has no affect on the final solutionbecause it was eliminated in both LHS and RHS.

Explicit relaxation for momentum and pressure equations. Explicit relaxationfor momentum and pressure equations is the regular technique, which is also calledunder-relaxation. In this form, the new solution is weighted by the old solutionusing the formula:

φ = (1 − α)φold + αφnew (64)

where φnew is the updated current solution and φold is the previous value. α is therelaxation factor. If α= 1, all previous values are ignored, and all updated values are



used. For convergence difficulties, low value of α will help. Automatic parametercontrol was adapted to eliminate the user’s input and adjustment effort. The controlmethod based on the convergence history identifies the convergence pattern via anintelligence control theory as proposed by Xie [16].

Solution strategy in fluid flow and heat transfer problemsBecause the governing equations are nonlinear, they must be solved iteratively.Picard successive substitution or Newton–Raphson iterative methods are used. InPicard successive substitution method, estimates of the solution variables (U, V,W, P, K, E, T, etc.) are substituted into the governing equations. The equations aresolved for new values that are then used as the estimates for the next pass. The globalsolver controller will either perform a fixed number of these global iterations orcheck for the convergence criterion. The controller will stop when either is reached.The convergence criterion is the level at which the specified variable’s residual normmust reach.Method A: For weak (one way) coupled flow and heat transfer problem

Loop (nonlinear) until convergenceSolve momentum equations by segregated schemeSolve pressure equations and velocity correction by segregated schemeEnd LoopSolve energy equations

Method B: For fully coupled flow and heat transfer problem

Loop (nonlinear) until convergenceSolve momentum equations by segregated schemeSolve pressure equations and velocity correction by segregated schemeSolve energy equationsEnd Loop

Method C: For weak (one way) coupled flow and heat transfer problem

Loop (nonlinear) until convergenceSolve coupled continuity, momentum equationsEnd LoopSolve energy equations

Method D: For fully coupled flow and heat transfer problem

Loop (nonlinear) until convergenceSolve coupled continuity, momentum equationsSolve energy equationsEnd Loop



Method E: For fully coupled flow and heat transfer problem

Loop (nonlinear) until convergenceSolve wholly coupled continuity, momentum and energy equationsEnd Loop

Owing to the importance of the segregated pressure–velocity solution schemeversus coupled scheme in most of the convective heat transfer problems, the majorityof computational efforts in heat transfer problems lie in solving the flow and pressurefields.

Scenario I demonstrates the decoupled flow and heat transfer, i.e., there areno temperature-dependent properties in the flow. We can first solve the isothermalflow (energy equation turned off) to yield a converged flow-field solution and thensolve the energy transport equation alone; this is a naturally segregated approach.Scenario II depicts the coupled flow and heat transfer (typically natural-convectionproblem); the common solution practice is to realize the flow–thermal couplingusing the nonlinear iteration until convergence. This is exactly the advantage of thevelocity–pressure “segregated scheme.”

Figure 5.4 shows the five solution schemes. Methods A and B are segregatedschemes described in this chapter, methods C and D are coupled momentumand pressure schemes, and method E is completely coupled velocity–pressure–temperature solution scheme. Method E is rarely used because of the bulk propertiesin global matrix by nature especially for large industry application. In any case, allmethods include solving energy equation.

For multiphysics system of equations to be solved by segregated manner, thesolution scheme is illustrated in Figure 5.4:

1. Velocity–pressure sequential solver (momentum-continuity)2. Turbulence subiteration3. Nonlinear iterations4. Time stepping iteration

At each time step, the nonlinear iterations are performed for the momentum-continuity, turbulence, front-tracking equation, and temperature. Subiterations ofturbulence transport equations are also used to accelerate the overall convergenceof the iterative process.

In more general and wider range multiphysics applications with structure, elec-tromagnetic, acoustics disciplines coupling with fluid flow and heat transfer, typicalapplication such as fluid–structure interaction (FSI), two-tier approaches may beapplied for the architecture with one unified simulation environment (workbench).In tier one, CFD package tightly integrate fluid flow, heat transfer, mass transfer, andacoustics components, and structure package tightly integrate linear dynamics andexplicit dynamics analysis components. In top level tier two, customized interpro-cess communication technology or general interprocess coupling tools like MPCCI



(I)

(IV)

(II)

Solve u momentum

Solve v momentum

Solve w momentum

Solve pressure andcorrect velocity

Solve flow

Initial condition(IC)

Solve turbulence kineticenergy

Solve eddy dissipation

Update mt

Solve T (heat transfer)

Solve front tracking(free surface problem)

While t tmax

(II)

(III)

Figure 5.4. General solution procedure of the segregated solver.

could be used for the communication among different independent packages suchas CFD, structure, electromagnetic analysis.

5.4.3 Data storage and block I/O process

Data storageThe FEM results in a system of linear equations containing a large number ofequations and unknown degree of freedom. A big amount of information requiresa clever data-keeping strategy to avoid whole matrix storage. Knowing that thesparse properties of global matrix, we are using data storing which is the so-calledskyline or column profile storage. Instead of storing every single matrix element,we could think of storing only the nonzero hits and corresponding position indexin the matrix assembly phase (Figure 5.5).

However, even with the use of such a reduced storage mode, the matrix is usuallyso large that it cannot be stored in core, so that it must be segmented in blocks andstored on low-speed disk storage.



X X

X X X

0

0 0

X X

X X X X

Figure 5.5. Skyline matrix structure.

Program Programbuffer

Operation systemarea

Systemcache

Outputfile

Figure 5.6. Block buffer I/O for disk writing.

Block I/O processImproving overall disk I/O performance can minimize both device I/O andactual CPU time. The techniques listed here can significantly improve FEA codeperformance.

The slowest process (routines) could be detected by using profiling tools such asAQTIME. After running a typical size of FEA model, the routines using most timein analysis, the top routines, and even lines are the processor bottleneck. These arethe targets to be improved for better performance.

“Block I/O” is a cost-efficient record buffer technique for large amount of datain the FEA processor. For writing, it saves a fixed number of element matrix infor-mation into buffer, and then writes the whole buffer into disk if the buffer is full,whereas reading is the reverse procedure. The buffer size is related to file system inoperation system and disk cache; we found that the optimal buffer size is 8 kbytesfor Windows XP WIN32 NTFS file system.

Here is an example for storage size calculation of one block element stiffnessmatrix with three-dimensional brick element (eight nodes) (Figure 5.6).Block size= 128 matrices* (8*8 entries/matrix) * 8 bits /entry= 65,536 bits= 65,536/8 bytes= 8,192 bytes



Table 5.4. Data structure for Block I/O

Part ID Element type Data mapping Block numbers

Simulation Part 1 Tetrahedral Block 1 (32 kB) 6 blocks

domain elements Block 2

(4 nodes) Block 3

Block 4

Block 5

Block 6 Block 6

Other Block 1 (32 kB) 5 blocks

hybrid Block 2

elements Block 3

(8 nodes) Block 4

Block 5 Block 5

Part 2 Tetrahedral Block 1 (32 kB) 7 blocks

elements Block 2

(4 nodes) Block 3

Block 4

Block 5

Block 6 (32 kB)

Block 7 Block 7

Other hybrid Block 1 (32 kB) 3 blocks

elements Block 2 (32 kB)

(8 nodes) Block 3 Block 3

The data structure for block buffer I/O is shown inTable 5.4. The fluid domain iscomposed of multiple parts, while each part is decoded as the a group of tetrahedralelements and the a group of hybrid elements, hybrid element could be the com-bination of any brick, pyramid, wedge elements which could be considered as thedegraded element from eight-node brick element. For each group, 128*4 matricesfrom tetrahedral element composed a block; each matrix has 4*4 element entries,which also has the block size of 8 kbytes.

Because the fluid domain is composed of parts and each part has a differentnumber of brick element and tetrahedral element while the number may not be theeven times of the optimal block element number, a special counter was used if therewere a remainder of element number over block size number.



Apart from the efficient use of record buffers and disk I/O, some other tips asthe following were used in the code implementation:

(a) Using unformatted files instead of formatted files.(b) Writing whole arrays.(c) Using pointer shallow copy (pointer) instead of the complete deep copy.

By using the above-mentioned comprehensive methods, the I/O performancecan be dramatically improved, but when compared to other operations, it becomestrivial.

5.5 Case Studies

This section shows two examples of conjugate heat transfer problems where bothsolid and fluid are involved in the heat transfer analysis. The results are obtainedby using finite element–based equal-order segregated solution scheme discussed inthis chapter, and the SUPG stabilization term is enforced in both fluid and thermalprocessors.

For simple geometry, the high cell Peclet number did not cause severe problemfor the simulation results. That is because the temperature gradient in the flowdirection happened to be small for the natural boundary condition applied at theexit. Consequently, the numerical diffusion introduced by the discretization alsobecomes small. For complex geometry, however, heat transfer occurs through bothconvection and diffusion in all directions. Whenever the convection effects arestrong, the numerical oscillation of the computed temperature would happen if theGalerkin formulation were used.

The reason for this numerical difficulty has been well described. The conclu-sion basically is that the Galerkin formulation is of the same nature as the centralfinite difference scheme. It does not consider the influence of the flow velocity onthe temperature distribution in a cell through its basic interpolation functions. Toovercome this difficulty, either an upwind scheme or optimized upwind scheme(SUPG) should be adopted for the convection term. In the upwind schemes, a biasis given to the upwind nodal temperature by modifying the interpolation functionsaccording to the local fluid flow velocity, both the direction and the magnitude.

5.5.1 Two-dimensional air cooling box

The first example shown in Figure 5.7 is a two-dimensional square-shaped electronicbox with airflow cooling. Two small copper blocks are heat sources with a constantvolumetric heat generation rate. The airflow comes in from an opening at the topedge near the left corner, and the outlet is located at the right side edge near thebottom corner. The flow rate and the temperature at the inlet are specified. Theinfluence of the inlet flow rate on the maximum temperature in the entire system,



Outlet

Inletcooling airT 25°C

Heatgenerationblocks

Convectionboundary

Figure 5.7. Schematic illustration for airflow cooling in two-dimensional electroniccase.

Table 5.5. Summary of parameters for airflow cooling system

Inlet velocity U 0.01 or 0.001 (m/s)

Inlet temperature Tin 25 (C)

Case dimensions 5 by 5 (m by m)

Heating block 1 (1, 1) to (2, 2) (x, y) – (x, y)

Heating block 2 (3, 3) – (4, 4) (x, y) – (x, y)

Convection film coefficient 0.2 (J/m2 s C)

Ambient temperature 25 (C)

Heat generation rate 20 (J/s/m3)

caused by the two heat sources, can be predicted. To verify the effects of the SUPGformulation, this case has also been analyzed using the Galerkin formulation.

The mesh used for the entire domain is 100×100, so the number of elementsfor this model is 10,000. Two element groups are used in order to specify heatgeneration rate for the two heat sources. The flow media is dry air and the twoheating blocks are copper. The edges of the two-dimensional box transfer heatto the ambient atmosphere through the natural convection. The convection filmcoefficient is assumed as a known value. A summary of the parameters used in thiscase study is given in Table 5.5.

The computed results for two different inlet flow velocities shown in Figure 5.7aand b are obtained if the simulation was carried out based on the Galerkin formu-lation. For (a) the cell Peclet number is about 20.0, while for (b) it is about 200.0.Temperature oscillations in some area of the flow domain may become severer forthe higher cell Peclet number case if SUPG stabilized term is not enforced.



Inlet velocity U 0.001 m/s(a) (b)

Inlet velocity U 0.01 m/s

Figure 5.8. Velocity isolines in cooling box.

The cooling effect can be observed by comparing Figure 5.8a and b. The resultwas achieved by using SUPG stabilized Galerkin discretization; the temperatureoscillations were completely eliminated by the optimized upwind SUPG scheme.

The higher inlet velocity makes more effective cooling result, the maximumtemperature for (a) is about 113.76C if inlet velocity is 0.001 m/s, while for(b) it is only about 64.50C if inlet velocity is 0.01 m/s. Relatively regular velocityand temperature patterns are maintained at smaller velocity since the flow stays inlaminar range. With increase in velocity, the flow becomes irregular and more vor-texes are developed, and the flow convection makes the corresponding temperaturepatterns.

Because the inlet velocity enters the box from the top side, the momentumlets the flow to hit the bottom heat generation block easily. This effect becomesmore obvious with increase in velocity, so that the temperature in the bottom heatgeneration block is lower than the top heat generation block even if the two blockshave the same amount of heat generation rate (Figure 5.9).

5.5.2 CPU water cooling analysis

A three-dimensional finite-element analysis is established for a CPU water coolingsystem, as shown in Figure 5.10. The bottom surface of the heat exchange block isactually in contact with the surface of a CPU chip.

A constant heat flux is therefore specified for this surface. It is assumed thatthe CPU has 130 W power dissipation and the heat is dissipated through the basesurface of the heat exchange block. Depending on the surface size and total power,the heat flux boundary condition is set as 38.5. Other external surfaces of the blockare exposed to the ambient atmosphere with a given convection film coefficient of 5.



25.000057 80.39644325.000000

25.000000

50.908382 45.792520

33.50671333.527300

25.014191

49.412597

55.70721264.494155

53.733615

26.881897

Temperature°C

Temperature°C

113.7605104.884596.0084487.1323878.2563369.3802760.5042251.6281642.7522133.8760525

64.5033260.5525156.601752.6508948.7000844.7792740.7984636.8476532.8968428.9460324.99522

113.755978

104.314131

76.834744

60.974225

47.29348362.560818

76.045629101.168845

96.056686

25.008885

Inlet velocity U 0.001 m/s(a) (b)

Inlet velocity U 0.01 (m/s)

Figure 5.9. Temperature contours in cooling box.

Mesh for the tube(enlarged view)

Figure 5.10. 3D model for a CPU water cooling system.

The tube wall is also modeled in this analysis, a part of the tube surface is exposedto the ambient atmosphere, and it has the same convection film coefficient as thatof the block surfaces. Both the block and the tube are made of the same material,which is aluminum in this case.

Cooling water with average velocity of 0.06 m/s at inlet goes through the tube toconvect the heat out of the system; the velocity profile in the water tube is calculatedby fluid flow analysis and then passed to thermal analysis as the convection term.The temperature at the entrance of the tube is specified as 25C in the analysis.The steady-state conjugated heat transfer problem is calculated by using the equal-order segregated solution scheme and SUPG stabilized term discussed earlier. Thisexample illustrates effective water cooling in the electronic system.



Temperature°C

61.7596158.0836554.4076950.7317347.0557743.3798139.7038436.0278832.3519228.6759625

Figure 5.11. Result contour for the CPU water cooling system.

The computed temperature contours are shown in Figure 5.11a. The temperatureranges from 25C to 61C. The numerical oscillation of the temperature solutionin traditional Galerkin formulation has been eliminated by using the SUPG term.This indicates that the SUPG formulation does provide a much better solutionover the Galerkin formulation, especially for convection-dominated heat transferproblems. The heat flux contour in the vicinity of the cooling flow inlet is shownin Figure 5.11b. It is noticed that the maximal heat flux is located in the vicinity ofthe inlet tube wall, which implies the maximal temperature gradient is there sincethe heat flux is proportional to the temperature gradient.

References

[1] Choi, H. G., and Yoo, J. Y. Streamline upwind scheme for the segregated formulationof the Navier-Stokes equation. Numerical Heat Transfer, Part B, pp. 145–161, 1994.

[2] Haroutunian, V., Engelman, M. S., and Hasbani, I. Segregated finite elementalgorithms for the numerical solution of large-scale incompressible flow problems,International Journal for Numerical Methods in Fluids, 17, pp. 323–348, 1993.

[3] Hill, D. L., and Baskharone, E.A.A monotone streamline upwind method for quadraticfinite elements, International Journal for Numerical Methods in Fluids, 17, pp. 463–475, 1993.

[4] Reddy, J. N., and Gartling, D. K. The Finite Element Method in Heat Transfer andFluid Dynamics, 2nd Edition, CRC Press, 2000.

[5] Rice, J. G., and Schnipke, R. J. A monotone streamline upwind finite elementmethod for convection-dominated flows, Computer Methods in Applied Mechanicsand Engineering, 48, pp. 313–327, 1985.

[6] Rice, J. G., and Schnipke, R. J.An equal-order velocity-pressure formulation that doesnot exhibit spurious pressure modes, Computer Methods in Applied Mechanics andEngineering, 58, pp. 135–149, 1986.

[7] Shaw, C. T. Using a segregated finite element scheme to solve the incompressibleNavier-Stokes equations, International Journal for Numerical Methods in Fluids, 12,pp. 81–92, 1991.



[8] Schunk, P. R., and Heroux, M. A. Iterative solvers and preconditioners for fully-coupled finite element formulations of incompressible fluid mechanics and relatedtransport problems, Sandia National Laboratories,Albuquerque, New Mexico, 87185–0835, 2001.

[9] Tezduyar, T., and Sathe, S. Stabilization parameters in SUPG and PSPG formulations,Journal of Computational and Applied Mechanics, 4(1), pp. 71–88, 2003.

[10] du Toit, C. G. A segregated finite element solution for non-isothermal flow, ComputerMethods in Applied Mechanics and Engineering, 182(3), pp. 457–481(25), 2000.

[11] Van Zijl, and du Toit, C. G. A SIMPLEST finite element solution of the incompress-ible Navier-Stokes equations, In: Proc. Symposium of Finite Element Method, SouthAfrica, 1992.

[12] Vellando, P. On the resolution of the viscous incompressible flow for various SUPGfinite element formulations, European congress on the Computational Methods in theApplied Sciences and Engineering – (ECCOMAS) 2000, Barcelona, 2000.

[13] Wang, G. A fast and robust variant of the simple algorithm for finite-element sim-ulations of incompressible flows, Computational Fluid and Solid Mechanics, 2,pp. 1014–1016, Elsevier, 2001.

[14] Wang, G. A conservative equal order finite element method using the collo-cated GALERKIN approach, 8th AIAA/ASME Thermo-physics and heat transferconference, St. Louis, Missouri, AIAA 2002-3206, 2001.

[15] Wansophark, N. Enhancement of segregated finite element method with adaptivemeshing technique for viscous incompressible thermal flow analysis, Science Asia.29(2), pp. 155–162, 2003.

[16] Xie, J. Equal-order segregated finite element formulation for flow and phase changeproblems, Doctoral Dissertations, University of Wisconsin, 2007.


III. Turbulent Flow Computations/LargeEddy Simulation/Direct NumericalSimulation


6 Time-accurate techniques for turbulent heattransfer analysis in complex geometries

Danesh K. TaftiDepartment of Mechanical Engineering, Virginia PolytechnicInstitute and State University, VA, USA

Abstract

With the large increase in computational power over the past two decades, time-dependent techniques in turbulent heat transfer are becoming more attractive andfeasible. The chapter describes the numerical and theoretical background to enablethe use of these methods in complex geometries.

Keywords: Heat exchangers, Large-eddy simulations (LES), Parallel computing,Turbomachinery, Unsteady CFD

6.1 Introduction

Heat transfer devices span a wide range of applications in the automotive, aerospace,electronics, power generation, and biomedical industries with characteristic lengthscales ranging from a few hundred microns to meters. The characteristic Reynoldsnumbers (Re) can range from low and largely viscous (Re< 10) to high and fullyturbulent (Re> 2,000). While predictions in the laminar regime do not pose signif-icant computational challenges, barring strong property variations, non-Newtonianflows, or significant body forces, predictions in the transitional and turbulentregime pose considerable challenges. In these flow regimes common techniquesemployed are direct numerical simulations (DNS), large-eddy simulations (LES),Reynolds-averaged Navier–Stokes (RANS) or unsteady RANS (URANS). DNS bydefinition requires that the full spectrum of turbulence down to the smallest dis-sipative scales be resolved in the calculation. As the Reynolds number increasesbeyond O(103 − 104) in wall-bounded flows, resolving the full spectrum becomesexceedingly expensive, which limits the application of DNS. LES extends its prac-tical range to Reynolds number of O(104 − 105) by resolving scales only up to theinertial range and modeling the smaller scales via a subgrid stress model. RANSextends this range further to very high Reynolds numbers by modeling the full spec-trum of turbulent scales and incorporating their time-averaged effect into the mean



momentum and energy balance. URANS is used in cases where external forcingis present at a given frequency or the flow has an intrinsic characteristic frequency(as in shedding of large-scale structures). By resolving only the large-scale time-dependent structures and modeling the smaller-scale turbulence, the required griddensity is still much less than other time-dependent methods.

The high computational cost of DNS and LES on the one hand and the predictionuncertainty of RANS on the other has led to the development and application ofhybrid RANS-LES methods, which have tried to combine the low cost of RANSwith the superior accuracy of LES. Out of this family of hybrid techniques, detachedEddy simulation (DES) has gained popularity since it was proposed by Spalart et al.[1] in 1997 for external flows with massive separation. DES involves sensitizationof a RANS model to grid length scales, thereby allowing it to function as a subgridstress model in critical regions.

All the above methods are valid in the range of flows encountered in heat transferdevices. For example, RANS and URANS and maybe DES are more readily appli-cable to system-level simulations, whereas DNS and LES are more appropriate forcomponent-level simulations.

The chapter focuses on the application of these techniques to complex turbulentflows highlighting the related theoretical and computational background from theauthor’s research experience. It starts with the transformed conservation equationsin general coordinate systems and their solution in a multiblock computationalframework. The nonstaggered grid pressure equation is related to its staggered gridcounterpart giving perspective to different approximations used in the literatureand the formulation finally used. A section is also devoted to the discretizationof the convection term and its effect on solution accuracy. This is followed by asection on subgrid modeling in LES and the DES method. Finally, the solution oflinear systems and parallelization strategies on modern computer architectures arediscussed. The chapter is closed by referring to some recent application of thesetechniques to complex heat transfer problems.

6.2 General Form of Conservative Equations

The continuum conservation of mass, momentum, and energy and the ideal gasequation of state are described by the following time-dependent, conservationequations (the superscript * is used to denote dimensional variables throughoutthis chapter):Mass conservation:

∂ρ∗

∂t∗+ ∇ · (ρ∗u∗) = 0 (1)

Momentum conservation:

∂(ρ∗u∗)∂t∗

+ ∇ · (ρ∗u∗u∗) = −∇P∗ + ∇ ·⎛⎜⎝µ∗( ∇u∗ + ∇u∗T )

−2

3µ∗( ∇ · u∗)I

⎞⎟⎠+ ρ∗g∗ (2)


Time-accurate techniques for turbulent heat transfer analysis 219

Energy conservation:

∂(ρ∗c∗pT ∗)

∂t∗+ ∇ · (ρ∗c∗

pu∗T ∗) = ∇ · (κ∗ ∇T ∗) +(∂P∗

∂t∗+ u∗ · ∇P∗

)+∗ + ρ∗g∗ · u∗

(3)

where

∗ =[µ∗( ∇u∗ + ∇u∗T ) − 2

3µ∗( ∇ · u∗)I

]: ∇u∗

is the viscous-dissipation term.Equation of state:

ρ∗ = P∗

R∗T ∗ (4)

Targeting applications in free and forced convection with variable fluid prop-erties, the equations are cast in nondimensional form by using the followingparametrization:

ρ = ρ∗

ρ∗ref

µ = µ∗

µ∗ref

κ = κ∗

κ∗refcp = c∗

p

c∗p_ref

x = x∗

L∗ref

u = u∗

U ∗ref

t = t∗U ∗ref

L∗ref

P = P∗ − P∗ref

ρ∗ref U ∗2

ref

T = T ∗ − T ∗ref

T ∗o

The reference temperature, T ∗ref and pressure P∗

ref are used for calculating all ref-erence property values, whereas T ∗

o is the nondimensionalizing temperature scale.The dynamic viscosity µ∗/µ∗

ref and thermal conductivity κ∗/κ∗ref variations with

temperature are represented by Sutherland’s law for gases (White [2]). While thesetwo quantities vary substantially with temperature, specific heat has a much weakerdependence. For example, viscosity and conductivity for air in the temperaturerange 240 to 960 K vary by more than 100%, whereas c∗

p only varies by 12%. Hence,assuming that c∗

p/c∗p_ref = 1, and that g∗ = g∗ey, the nondimensional equations are

written as:Mass conservation:

∂ρ

∂t+ ∇ · (ρu) = 0 (5)


∂(ρu)∂t

+ ∇ · (ρuu) = −∇P + 1

Re∇ · (µ( ∇u + ∇uT ) − 2

3µ( ∇ · u)I)

+ RaRe2Pr

ρ

(T )ey

(6)




∂(ρT )∂t

+ ∇ · (ρuT ) = 1

RePr∇ · (κ ∇T ) + Ec

(∂P

∂t+ u · ∇P

)+ Ec

Re

+(

Ec

Fr

)ρ ey · u

(7)

Equation of state:

ρ = ρ∗ref U ∗2

ref P + P∗ref

R∗ρ∗ref (T · T ∗

o + T ∗ref )

(8)

where T in equation (6) is the characteristic nondimensional temperaturedifference that drives the buoyant flow,

Re = ρ∗ref U ∗

ref L∗ref

µ∗ref

, Pr = µ∗ref c∗

p_ref

κ∗ref, Ra = g∗β∗(T ∗)L∗3

ref

α∗ref ν

∗ref

Ec = U∗2ref

c∗p_ref T ∗

o, and Fr = U ∗2

ref

g∗L∗ref

specify the Reynolds, Prandtl, Rayleigh, Eckert, and Froude number, respectively,and β∗ is the thermal expansion coefficient. The reference velocity U ∗

ref is selectedbased on the dominating convection force. In forced convection, the referencevelocity is specified from the driving flow, e.g., the free-stream velocity in externalflow. In the case of pure natural convection, the reference velocity is calculatedfrom the characteristic length and the driving temperature difference, as defined bythe following (dimensional) equation:

U ∗ref =

√g∗L∗

refβ∗T ∗ (9)

In equation (6), the nondimensional coefficient of the gravitational body forceterm, Ra/(Re2Pr), determines the gravitational effect on the flowfield. In pure natu-ral convection, the term takes on a value of unity after substitution of the referencevelocity (equation (9)), into the Reynolds number, whereas in forced convection,when Re>>Ra, the coefficient would be negligibly small. In mixed convection,both the specified driving temperature difference and reference convective veloc-ity determine the relative magnitude of the gravitational source term and hence ofnatural versus forced convection.

6.2.1 Incompressible constant property assumption

Low-speed incompressible flow with constant fluid properties results in a numberof simplifications in the conservation equations. One of the main numerical con-sequences of this assumption is the decoupling of density from pressure and the



implicit assumption that pressure waves travel at infinite speed relative to the flowvelocities (Mach number, Ma → 0).The Eckert number, Ec = Ma2(γ − 1) → 0, andthe pressure work term in equation (7) makes a negligible contribution to energyconservation and can be neglected. In addition it is also customary to neglect theviscous dissipation term (Ec/Re → 0) and the potential energy term (Ec/Fr → 0).However, it is noteworthy that for very low Reynolds number highly viscous flowconditions might exist at which the viscous dissipation term can no longer beneglected. After simplifications, the nondimensional equations take the form:

ρ = ρ∗

ρ∗ref

= 1; µ = µ∗

µ∗ref

= 1; κ = κ∗

κ∗ref= 1; cp = c∗

p

c∗p_ref

= 1

x = x∗

L∗ref

; u = u∗

U ∗ref

; t = t∗U ∗ref

L∗ref

; P = P∗ − P∗ref

ρ∗ref U ∗2

ref

T = T ∗ − T ∗ref

T ∗o

Mass conservation:

∇ · u = 0 (10)


∂u∂t

+ ∇ · (uu) = −∇P + ∇ ·(

1

Re∇u

)+ gey (11)


∂T

∂t+ ∇ · (uT ) = ∇ ·

(1

Re Pr∇T

)(12)

where g = 1/Fr = g∗L∗ref /U

∗2ref .

For small temperature and density differences (ρ∗ = ρ∗ref +ρ∗) with other

properties held constant, the Boussinesq approximation can be accommodated inequation (11) by replacing the last term with gey(1 − T ) if T ∗

ref = T ∗0 .

Fully developed assumptionIn a number of applications the geometry lends itself to considerable computationalsimplification by considering a single period of a repeating feature in space such asin an array of fins and in ducts with repeating roughness elements. This is done byinvoking a fully-developed assumption in the flow direction, thus allowing simula-tion of a characteristic unit of the flow. This formulation was developed for steadyflow by Patankar et al. [3] for both a constant heat flux and a constant tempera-ture boundary condition. Because of the usefulness of this formulation for detailedanalysis of flow physics and heat transfer with time-dependent techniques, it isrepeated here for a constant heat flux boundary condition in the context of unsteadyflow and heat transfer under conditions of a fixed pressure gradient and a develop-ing flow field. Under these conditions, the wall friction velocity u∗

τ =√τ∗weq/ρ∗ is



used as the reference velocity to nondimensionalize the governing equations. For afully-developed flow a mean momentum balance in the flow direction (x) gives thefollowing relationship:

U ∗ref = u∗

τ =√τ∗weq/ρ∗ =

√(−P∗

x /L∗x )(D∗

H/4ρ∗) (13)

where τ∗weqis an equivalent mean wall shear which also includes form losses in the

domain, P∗x is the mean pressure drop in the flow direction across the computa-

tional domain length of L∗x , and D∗

H is the hydraulic diameter. The characteristictemperature scale is defined as T ∗

o = q∗wL∗

ref /κ∗, where −q∗

w is the applied wallheat flux. Since both pressure and temperature have a x-directional dependence,the assumed periodicity of the domain in the streamwise or x-direction requiresthe mean gradient of pressure and temperature to be isolated from the fluctuatingperiodic components as follows:

P∗(x, t) = P∗ref − β∗x∗ + p∗(x, t)

T ∗(x, t) = T ∗ref + γ∗(t) · x∗ + θ∗(x, t)

(14)

where β∗ = −P∗x /L

∗x is the mean pressure gradient, p∗ is the periodic pressure

fluctuations, γ∗ is a time-dependent temperature gradient, and θ∗ is the fluctuatingor periodic temperature component. Nondimensionalizing equation (14) gives

P(x, t) = −βx + p (x, t)

T (x, t) = γ (t)x + θ (x, t)(15)

where β= 4/DH from equation (13).Substituting equation (15) in the nondimensional conservation equations (10)–

(12) gives the following modified momentum and energy equations:Momentum conservation:

∂u∂t

+ ∇ · (uu) = −∇p + ∇ ·(

1

Re∇u

)+ gey + βex (16)


xdγ

dt+ ∂θ

∂t+ ∇ · (uθ) = ∇ ·

(1

Re Pr∇θ

)− γux (17)

with modified boundary conditions

φ (x, t) = φ (x + Lx, t),φ = u, p, and θ

and

−∇θ · n = −1 + γex · n



14

12

10

8

6

0 5 15 2010Time

5.5 × 10−4

5.4 × 10−4

5.3 × 10−4

5.2 × 10−4

5.1 × 10−4

5.0 × 10−4

Time-dependent formulation

Nuδ

γ

γ

Nu δ

Quasi-steady formulation

Figure 6.1. Time evolution of γ using the unsteady and quasi-steady procedure.The time-varying Nusselt number calculated on the two walls isidentical between the two formulations.

The time evolution of γ is obtained by a global integration of equation (17).Under impermeable wall conditions

VLx

2

dγ

dt+

∫V

∂θ

∂tdV = −

∫S

1RePr

(−1 + γex · n)dS − γQxLx (18)

where V is the fluid volume in the computational domain, S is the heat transfersurface area, and Qx is the x-directional flow rate. The left-hand side represents thetime-dependent terms and under assumed quasi-steady conditions the expressionfor γ is given as:

γ = S

RePrQxLx(19)

which represents an instantaneous energy balance between the heat added to thedomain at heat transfer surfaces and that removed volumetrically by the (sink)term in the energy equation (−γux). The reduction of equation (18) to (19) uses∫s

ex · nds = 0 for closed heat transfer surfaces.Figure 6.1 plots the time variation of γ in the heat transfer calculations of tur-

bulent channel flow at Re = Reτ = 180 (based on friction velocity uτ and channelhalf-width δ). In one calculation, γ is calculated based on the time-dependent evolu-tion using equation (18) and in the other it is based on the quasi-steady assumptiongiven by equation (19). The temperature field is initiated at Time = 0 with constantheat flux boundary conditions at the channel walls after the turbulent velocity field



is fully developed. Whereas the quasi-steady formulation is affected by the instan-taneous variations in the flow rate, the unsteady formulation exhibits near constantvalues of γ . In spite of these differences, the instantaneous average Nusselt num-bers based on channel half-width on the two channel walls are identical between thetwo calculations. From these and other similar results, the quasi-steady assumptionin calculating γ can be deemed adequate for most cases when there is no extremedriving unsteadiness in the applied wall heat flux or in the applied pressure gradient.

6.2.2 Modeling turbulence

For turbulent flows, the appropriate ensemble averaged equations for RANS or fil-tered equations for LES are constructed in physical space to obtain the Reynoldsstress terms and the subgrid terms, respectively. For a RANS formulation theReynolds stresses to be modeled take the form:

[τ]; τij = u′iu

′j (20)

whereas the subgrid stresses take the form:

[τ]τij = uiuj − uiuj + u′′i uj + uiu′′

j + u′′i u′′

j Or τij = uiuj − uiuj (21)

where the overbar represents the Reynolds-averaging operator for RANS and thefiltering operation for LES, respectively. In filtering the LES equations it is assumedthat the filter is commutative, i.e., ∂f /∂x = ∂f /∂x. Similarly, u′ represents the tur-bulent fluctuating quantities for RANS and u′′ the unresolved subgrid velocities.In equation (21) the underlined term is the Leonard stress term, which denotesinteractions between the resolved scales contributing to subgrid terms, followed bythe interaction between grid and subgrid scales, and between subgrid scales only.Within the framework of a linear isotropic eddy-viscosity model for incompressibleflow, both stress tensors are modeled as:

τaij = τij − 1

3δijτkk = − 2

RetSij (22)

where 1/Ret is the nondimensional turbulent eddy viscosity for RANS andthe subgrid eddy viscosity for LES and Sij = 1/2(∂ui/∂xj + ∂uj/∂xi) or [S] =1/2

( ∇u + ∇uT)

is the mean or filtered strain rate, respectively. The energy

equation is treated analogously to the momentum equations to give

τj = − 1PrtRet

∂T

∂xj= − 1

PrtRet

∇T (23)

where Prt is the turbulent Prandtl number. For convenience, in the rest of the chapterthe overbar notation is dropped in the rest of the chapter being mindful of the factthat the same set of equations are used for RANS and LES, which are ensembleaveraged for RANS and filtered for LES.



6.3 Transformed Equations in Generalized CoordinateSystems

For generalization to complex geometries, equations (10–12) are mapped fromphysical coordinates (x) to logical/computational coordinates (ξ) by a boundaryconforming transformation x = x(ξ), where x = (x, y, z) and ξ= (ξ, η, ζ). Based onthe developments in Thompson et al. [4], a set of covariant basis vectors are definedwhich are tangents to the coordinate curves ξi

ai = xξi , i = 1, 2, 3

Also defined is the symmetric covariant metric tensor G, whose elements are

gij = ai · aj = gji

where√

g =√det |g| = a1 · (a2 × a3) is the Jacobian of the transformation, which

is the ratio of element volume in physical space to that in computational space.Defined further are a set of contravariant vectors that are perpendicular to

coordinate surfaces, ξi = const.,

ai = ∇ξi = 1√g

(aj × ak ), i = 1, 2, 3 and (i, j, k) are cyclic

where√

gai is a measure of the area of the coordinate surface ξi = const. Thecorresponding symmetric contravariant metric tensor with elements:

gij = ai · a j = g ji

is then defined.Under these transformations the conservative divergence operator is defined as

∇ · A = 1√g

∂

∂ξi(√

gA · ai), i = 1, 2, 3

the conservative gradient operator as

∇φ = 1√g

∂

∂ξi(√

gaiφ), i = 1, 2, 3

and the conservative Laplace operator as:

∇ · ∇φ = 1√g

∂

∂ξi

(1√g

∂

∂ξk(√

ggikφ))

, i, k = 1, 2, 3



Finally, the transformed governing equations (10–12) are written as:Mass conservation:

∂

∂ξj(√

gU j) = 0, (24)


∂

∂t(√

gui) + ∂

∂ξj(√

gU jui) = − ∂

∂ξj(√

g(aj )iP)

+ ∂

∂ξj

((1

Re+ 1

Ret

)√gg jk ∂ui

∂ξk

)+ √

gSui

(25)

Energy equation:

∂

∂t(√

gT )+ ∂

∂ξj(√

gU jT ) = ∂

∂ξj

((1

PrRe+ 1

PrtRet

)√gg jk ∂T

∂ξk

)+√

gST (26)

Here√

gU j = √g(a j)kuk is the contravariant flux vector where (a j)k denotes the

kth element of the vector a j , and ui the Cartesian velocity basis vector. Sui and ST

are additional source terms in the momentum and energy equation, respectively.For the fully-developed formulation, while the mass continuity equation remains

unchanged, the momentum and energy equation take the form:Momentum conservation:

∂

∂t(√

gui) + ∂

∂ξj(√

gU jui) = − ∂

∂ξj(√

g(a j)i p) + ∂

∂ξj

((1

Re+ 1

Ret

)√gg jk ∂ui

∂ξk

)+ √

gβex + √gSui (27)

Energy equation:

∂

∂t(√

gθ) + ∂

∂ξj(√

gU jθ) = ∂

∂ξj

((1

PrRe+ 1

PrtRet

)√gg jk ∂θ

∂ξk

)− √

gγux − √gx∂γ

∂t+ √

gST (28)

where Re = Reτ is the Reynolds number based on the friction velocity u∗τ

(Equation 13).

6.3.1 Source terms in rotating systems

Equations (24–26) are the base incompressible conservation equations in a gen-eralized coordinate system. Additional terms due to body forces can be includedin the source terms Sui and ST . In noninertial systems as that encountered in theinternal cooling of turbine blades additional Coriolis and centrifugal force termsare included in nondimensional form:

√gSui = −2

√g Rokum ∈ikm −√

g ∈ilm∈jlk RojRomrk (29)



where Roi is the rotation number given by ω∗i L∗

ref /U∗ref , ω∗

i is the component ofangular velocity in the i-direction, rk is the radial component from the axis of rota-tion in the k-direction, and ∈jlk is the permutation tensor. In the case of orthogonalrotation about one axis (say the z-axis; ω∗

1 = ω∗2 = 0), equation (29) reduces to

√gSu1 = 2

√gRo3u2 + √

gRo23r1;

√gSu2 = −2

√gRo3u1 + √

gRo23r2√

gSu3 = 0(30)

Coriolis forces have a preferential effect on turbulence, augmenting or attenu-ating it depending on the orientation of the rotational axis with respect to theprimary strain rate in the flow. The preferential effect has a profound impacton heat transfer coefficients, and fluid temperatures can vary substantially in thedomain making the density differences nontrivial. The density differences give riseto differential centrifugal forces which are aptly referred to as centrifugal buoy-ancy forces. This is usually accommodated by using the Boussinesq approximationρ∗/ρ∗

ref = −T∗/T ∗ref to give

√gSui = −2

√gRokum ∈ikm −√

g ∈ilm∈jlk RojRomrk

(1 − T

T ∗0

T∗ref

)(31)

where the second term in the parenthesis represents the centrifugal buoyancy term.For developing and fully-developed flow T ∗

ref = T ∗in, and T ∗

0 = (T ∗w − T ∗

in) or T ∗0 =

q∗ref L∗

ref /κ∗ for constant temperature and constant heat flux boundary conditions,

respectively.

6.4 Computational Framework

The transformed equations can either be solved on a staggered or nonstaggered grid.Here the equations are discretized and solved on a nonstaggered grid using a finite-volume approach. The dependent variables u, v, w, P, or p and T or θ are calculatedat the cell center, while the contravariant flux vector, which is representative of thenormal flux to a coordinate surface, is defined and calculated at the cell faces,

√gU i

at ξι= const. Typically, the grid metrics are calculated using appropriate second-order central (SOC) difference approximations. If high-order accuracy is desired,high-order approximations are used to calculate the grid metrics.The covariant basisvectors (a1, a2, a3), the Jacobian

√g, the contravariant basis vectors (a1, a2, a3), the

diagonal elements of the covariant metric tensor (g11, g22, g33), and off-diagonalelements of the contravariant metric tensor (g12, g23, g13) are calculated and storedat the cell center. Further, to calculate the contravariant fluxes at the cell faces thecontravariant basis vectors and diagonal elements of the contravariant metric tensorare calculated and stored at the appropriate cell face (ai, gii at ξi = const).

The above equations are discretized and solved in a multiblock framework. Thisis accomplished by introducing an overlap region or ghost cells at interblock faces.Typically, the overlap region is dictated by the order of discretization used in the



η+ Faceζ+ Face

Axis 2

Axis 2

η

ζζ

ζ

Axis 1

Axis 1

Figure 6.2. An example of unstructured block topology. East face (ξ+) of one blockexchanges information with the north face (η+) of an adjoining block.The axes are also arbitrarily oriented with each other.

domain – for second-order discretizations, a single ghost cell region is needed atan interblock boundary; for second-order upwind approximations and fourth-ordercentral approximations, the ghost cells increase to two and so on. Obviously, theexpected increase in accuracy has to be balanced against the increased cost of thescheme itself, to which updating the overlap regions would add to, particularly in aparallel distributed computing framework.

While each block has a structured mesh, the interblock topology can eitherbe structured or unstructured. In a structured topology, a ξ+ block face can onlyinterface with an adjoining ξ− block face with coordinate axes perfectly alignedwith each other on the two faces. In an unstructured topology, this restriction isrelaxed and a ξ+ block face can interface with any of the six adjoining block faceswith any axes orientations. Figure 6.2 shows such an example in which the ξ+ faceof one block interfaces with η+ face of another block.

In this instance, data transferred from one face to the other has to be sortedand realigned with the receiving coordinate system. The sorting operation isabstracted as:

A|ξ+ = [S]

A|η+ (32)

where

A is the data-vector (of size equal to the number of nodes in the surface) tobe transferred from one face to the other and [S] is a symmetric sorting matrix.

If the elements of

A have a directional dependence on the transformed coordi-nates such as components of the covariant vectors xξ, xη, xζ or other derivatives w.r.t.ξ, η, and ζ, an additional transformation is performed. For example in Figure 6.2,φη|ξ+face = −φξ|η+face. A general transformation is represented as:

⎡⎣φξφηφζ

⎤⎦ξ+face

= [S][T ]

⎡⎣φξφηφζ

⎤⎦η+face

(33)



where [T] is a 3 × 3 transformation matrix. For the example in Figure 6.2

[T ] =⎡⎣ 0 1 0

−1 0 00 0 1

⎤⎦An additional feature that is useful is the capability to handle nonconforming ornonmatching block interfaces, i.e., when there is no one-to-one match between nodelocations in two adjoining block interfaces.This capability is useful in zonal refiningor coarsening of meshes from one block to another. In such cases, in addition tothe sorting matrix, an interpolation matrix is constructed. Consider the η+ face inFigure 6.2 to have n total nodes, which need to be transferred to ξ+ face with mnodes. Then a further generalization of equation (32) can be written as:

A|ξ+m×1 = []m×n[S]n×n

A|η+n×1 (34)

where [] is the m × n interpolation matrix.The interpolation operator should at a minimum have the same order of accuracy

as the discrete approximations in the code. Bi-linear interpolation is the simplestgiving close to second-order accuracy. Higher-order methods based on quadraticor cubic splines can also be used but at a higher cost [5]. During interpolation it isalso important to conserve the mass, momentum, and energy fluxes moving acrossthe interface. Some elements of this are explained in Section 5.3.

Figure 6.3 shows results from a zonal mesh used for the simulation of turbulentchannel flow at Reτ = 180. The results are compared to the DNS data of Kim et al.[6] (KMM). Turbulent channel flow in spite of its geometrical simplicity embod-ies all the essential characteristics of wall-bounded turbulence and is an ideal testcase for numerical developments in the simulation of turbulent flow. The zones aredemarcated at y+ = 50 and 100, with resolutions of 64 × 14 × 80, 48 × 9 × 64, and32 × 9 × 48 cells, respectively, for Lx = 2π and Lz =π and half-channel height fora total resolution of 226,304 cells on a six-block mesh. Because of the relativelysmall-scale turbulence present in the flow, it provides a stringent test case for thetreatment of nonmatching boundaries. The results are compared to a single zonemesh of 64 × 64 × 64 or 262,144 cells. Bi-linear interpolations are used at blockboundaries, together with Dirichlet boundary conditions for pressure with the con-servation of integrated mass flux and pressure gradients across zonal boundaries.The overall prediction accuracy is satisfactory for the zonal decomposition, but peakvalues are slightly underpredicted compared to the single zone mesh. In addition, atthe zonal boundary y+ = 50, the normal and shear stresses exhibit discontinuitiesin slope.

This can be attributed to the highly energetic small-scale eddies in this region andthe resulting interpolation errors. Hence, unless high-order interpolations are used,nonmatching boundaries should be used in regions where the flow has relativelylarge-scale features that can be captured relatively well on the grid [7].



Z X

Y

(a)

(b)

2.6

2.4

2.2

Symbols DNS (KMM)

Zonal mesh

Single mesh

1.8

1.6

1.4

1.2

u rm

s, v

rms,

wrm

s

0.8

0.6

0.4

0.2

100 101

y+102

vrms

wrms

urms

0

1

2

0.8

0.6

0.4

turb

ulen

t she

ar s

tres

s (−

u′v

′)

DNS (KMM)

Zonal meshSingle mesh

0.2

10−1 100 101

y+ 1020

Symbols

(c)

Figure 6.3. (a) Zonal mesh in x–y plane; (b) rms quantities in wall coordinates;(c) turbulent shear stress distribution in wall coordinates. Zonalinterfaces are at y+ = 50 and 100.



6.5 Time-Integration Algorithm

A number of algorithms are available for time integration of the conservativeequations. The most common methods for incompressible flow are the so-called“pressure” based methods, which solve for pressure by formulating a pressureequation using the mass continuity equation. Typically within this framework, themomentum equations are integrated in time using some guess for the pressure gradi-ent (from a previous iterate or time step) to give an intermediate velocity field, whichis then made to satisfy mass continuity by correcting with the calculated pressurefield. Whereas for steady-flow calculations, algorithms based on the semi-implicitpressure-linked equations (SIMPLE) method are quite popular (Patankar [8];Versteeg and Malalasekera [9]), for unsteady solutions the fractional step approachfinds widespread use (Ferziger and Peric [10]). Whereas methods based on theSIMPLE algorithm solve the full set of nonlinear equations iteratively at each timestep and hence technically do not have any time step restrictions, methods basedon fractional step can be fully explicit with time step limited by both convectiveCourant–Frederichs–Levy (CFL) condition and viscous limits (Chorin [11]), semi-implicit in which only the viscous terms are treated implicitly (Kim and Moin [12]),and fully implicit in which both convection and viscous terms are treated implic-itly (Choi and Moin [13]). Although time step limitations in the fractional stepapproach can be an impediment to efficient solution, for unsteady turbulent flowssmall time steps are indeed desirable to resolve the small turbulent time scales. Infact a balanced spatio-temporal discretization dictates a CFL of O(∼1) and largetime steps can have the same undesirable effect on the solution as a coarse spatialgrid or a dissipative spatial discretization.

Here three algorithmic implementations within the fractional step approach aregiven. In the predictor step an intermediate velocity field ui is computed explicitly,semi-implicitly, or implicitly by neglecting the effect of pressure gradient. In theexplicit treatment, a second-order accurate in time Adams-Bashforth scheme isused to approximate both the convective and viscous terms. In the semi-implicittreatment, which is primarily used for low Reynolds number flows, the viscousterms are treated implicitly by a second-order Crank–Nicolson approximation. Inthe implicit treatment, both convective and viscous terms are treated by a Crank–Nicolson approximation. In all three methods the convection terms are linearizedby using the contravariant fluxes at time level n or n−1. Additionally, built into thealgorithm is the explicit conservation of surface-integrated mass, momentum, andenergy at nonmatching block interfaces, which yield conserved variables 〈√gU j〉,〈un

i 〉, and 〈T n〉, respectively, where 〈〉 denotes conserved variables. The energyequation needs only the predictor step to obtain T n+1 and takes a form similar tothe equations (35)–(37) below.

6.5.1 Predictor step

First, momentum and energy fluxes across nonmatching interfaces are conservedto obtain the conserved quantity 〈un

i 〉 or 〈T n〉.



Explicit formulation:

ui − 〈uni 〉

t= 3

2H n

i − 1

2H n−1

i (35)

where

Hi = − 1√g

∂

∂ξj(〈√gU j〉〈ui〉) + 1√

g

∂

∂ξj

((1

Re+ 1

Ret

)√gg jk ∂〈ui〉

∂ξk

)+ Sui

Semi-implicit formulation:

ui − 〈uni 〉

t= 3

2H n

i − 1

2H n−1

i + 1

2

1√g

∂

∂ξj

[(1

Re+ 1

Ret

)√gg jk

(∂ui

∂ξk+ ∂〈un

i 〉∂ξk

)](36)

where Hi = − 1√g∂∂ξj

(〈√gU j〉〈ui〉) + Sui

Implicit formulation:

ui − 〈uni 〉

t= −1

2

1√g

∂

∂ξj

(√gU jui + 〈√gU j〉〈un

i 〉)

+ 1

2

1√g

∂

∂ξj

[(1

Re+ 1

Ret

)√gg jk

(∂ui

∂ξk+ ∂〈un

i 〉∂ξk

)]+ Sui

(37)

where√

gU j = 2√

gU jn − √gU jn−1

is a linearized estimation of√

gU j.The superscript ∼ denotes the intermediate field, 〈〉 denotes quantities based on

conserved fluxes at nonmatching boundaries, and n the present time level. In thesemi-implicit and implicit formulation, at inhomogeneous boundaries ui = un+1

iin the momentum equations and T = T n+1 are used in the energy equation. Thesemi-implicit and implicit splitting of the momentum equations introduce an errorof O(t2), which is of the same order as the temporal discretization.

6.5.2 Pressure formulation and corrector step

After obtaining the intermediate Cartesian velocity field, the continuity equation isused to derive the pressure equation, which is then solved to obtain the pressurefield at time (n + 1)t. The calculated pressure field is then used to correct theintermediate velocity field to satisfy the divergence-free condition. On a staggeredgrid, this procedure is straightforward since the intermediate velocities are availableon the cell face of the pressure node and there is no ambiguity in satisfying discretecontinuity. On the other hand, on a nonstaggered grid, interpolations of nodalvelocities and pressure gradient terms are necessary at the cell face. This leads toa discrete pressure Laplacian, which does not exhibit good ellipticity as dictatedby the continuous Laplace operator and leads to the classic problem of grid scale



i + 1i − 1 i

i + 1/2

i

i + 1/2i − 1/2

Staggered

Nonstaggered

i − 1/2

i − 1

u-velocity

i + 1

Figure 6.4. One-dimensional staggered and nonstaggered grids.

pressure oscillations (Tafti et al. [14]). The alleviation or elimination of this problemis an important issue on nonstaggered grids and is dealt with in some detail here. Tounderstand it better, a general framework is developed in which the nonstaggeredgrid approximations are written in terms of their staggered grid counterparts. Thisframework is then used to elucidate on some of the techniques used in the literatureto approach the staggered grid formulation.

Figure 6.4 shows a one-dimensional (1-D) Cartesian staggered and nonstag-gered grid, with the locations of velocity and pressure. On a staggered grid, thenondimensional x-momentum equation is solved at i ± 1/2 location as follows:(

∂u

∂t

)i+1/2

= f (u) − pi+1 − pi

(38)

where f (u) are the convection and viscous terms. Here, the pressure gradient termis approximated by a compact gradient operator. On the nonstaggered grid, thex-momentum equation with a SOC difference approximation for the pressuregradient at location i is discretized as follows:(

∂u

∂t

)i= f (u) − pi+1 − pi−1

2(39)

In this case, the pressure gradient is approximated by the noncompact form ofthe gradient operator. By using Taylor series approximations, we can rewrite thisequation in a compact gradient operator form and an “error” term as follows:(

∂u

∂t

)i= f (u) − pi+1/2 − pi−1/2

− 2

8

(∂3p

∂x3

)i+ . . . (40)

In essence, the nodal momentum equation on a non-staggered grid has an additionalthird derivative pressure term implicitly included in its formulation which can be



written as: (∂u

∂t

)ns

=(∂u

∂t

)s

− a

ε1

(∂a+1p

∂xa+1

)+ . . . (41)

where the superscript ns refers to nonstaggered and s refers to staggered grid. Inthis case a = 2, and ε1 = 8.

In deriving the pressure equation, the 1-D equivalent of the discrete continuityequation is of the form:

ui+1/2 − ui−1/2

= 0 or

1

[(∂u

∂t

)i+1/2

−(∂u

∂t

)i−1/2

]= 0 (42)

In the staggered grid system, ui+1/2 and ui−1/2 are available at the cell faces fromthe x-momentum equation. However, in the nonstaggered grid, these velocities haveto be interpolated to the cell faces. Using second-order symmetric interpolationsfor (∂u/∂t)i+1/2 and equation (39) gives

(∂u

∂t

)i+1/2

= 12

(f (u)i+1 + f (u)i) − pi+2 + pi+1 − pi − pi−1

4(43)

Using Taylor series expansions for f (u) about the staggered node i + 1/2, andexpressing the pressure gradient term in terms of its compact form (pi+1 − pi,),equation (43) can be written as:

(∂u

∂t

)i+1/2

= f (u)i+1/2 − pi+1 − pi

+

b

ε2

(∂bf (u)

∂xb

)i+1/2

− c

ε3

(∂c+1p

∂xc+1

)i+1/2

+ . . .(44)

or in general

(∂u

∂t

)ns

=(∂u

∂t

)s

+ b

ε2

(∂bf (u)

∂xb

)− c

ε3

(∂c+1p

∂xc+1

)+ . . . (45)

where b = 2, ε2 = 8, c = 2, and ε3 = 4. Equation (45) gives a general expression forthe equivalent cell face momentum equation used on a nonstaggered grid in termsof its staggered grid counterpart to construct the pressure equation. The expressionincludes two additional terms, one of which is a result of the cell face interpolation,and the other, as in the expression for the nodal velocity, is a result of the noncompactform of the gradient operator. It is to be noted that the order of the interpolationerror term can in fact be reduced by using high-order interpolations. Zang et al.[15] have used a QUICK type upwind biased approximation to interpolate the cellface values which gives b = 3 and ε2 = 24 in equation (45). However, although the



magnitude of the error term introduced by the noncompact form of the pressuregradient can be reduced (Tafti et al. [14]), the order of the error always remainsO(2) with the third derivative of pressure.

Finally, using the 2-D equivalent forms of equations (45) and (42) a modifiedpressure equation in two dimensions is derived as:

∇2ps + cx

ε3

(∂c+2p

∂xc+2

)+ c

y

ε3

(∂c+2p

∂yc+2

)+ . . . = 1

t(∇ · u)s

+bx

ε2

(∂b+1f (u, v)

∂xb+1

)+ b

y

ε2

(∂b+1g(u, v)

∂yb+1

)+ . . .

(46)

where ∇2ps is a compact Laplacian and (∇ · u)s is the compact divergence operatoras obtained on a staggered grid. The constants b, c, ε2, and ε3 have the samevalues as in equation (45). It is clear that the extra terms in the modified pressureequation introduce high frequency errors of O(2) into the pressure field, whichcontribute to the grid scale pressure oscillations. The elimination of the spuriouspressure terms in equation (46) has been researched extensively. For example,Sotiropoulus and Abdallah [16] use regularizing terms in the pressure equation (of

type − ε4

2 ∂4p∂x4 , 0 ≤ ε≤ 1) which partially or exactly cancel the spurious terms in

equation (46). In a 2-D-driven cavity they found that ε= 0.1 was sufficient to obtainsmooth pressure fields. However, by arbitrarily “regularizing” the pressure equation,discrete continuity is not exactly satisfied. In order to overcome this deficiency,Tafti et al. [14] used equation (42) for discretizing the divergence operator in thecontinuity equation, and directionally biased the pressure gradient operator to obtaina discrete Laplacian that exhibited better ellipticity (L23 Laplacian in reference[14]). This Laplacian was found to eliminate pressure oscillations in a 2-D-drivencavity and also satisfied discrete continuity. For this approximation, equation (39)took the equivalent form:

(∂u

∂t

)i= f (u) − −pi+2 + 6pi+1 − 3pi − 2pi−1

6(47)

resulting in a = 2, and ε1 = −24 in equation (41) for the nodal momentum equation,b = c = 2, and ε2 = 8 and ε3 = 12 in equations (45) and (46). Hence, the direction-ally biased pressure gradient approximation was successful in reducing the errorin the spurious pressure term by a factor of 3; however, as noted previously, theorder of the error term remained the same. A limitation of this method is that thediscrete Laplacian has a large stencil and hence makes implementation in a generalcoordinate frame quite cumbersome.

An additional criterion that needs to be satisfied when solving the pressureequation is the integrability constraint. It states that the discrete pressure Poisson



equation should satisfy Gauss divergence theorem, i.e.,∫V

D · (Gp)dV =∫S

(Gp) · nds

and ∫V

D · udV =∫S

u · ndS (48)

where D and G are the discrete divergence and gradient operators, u is the nondi-mensional velocity vector, n is the outward pointing unit normal at the surface sbounding the computational domain of volumeV . This simply states that the dis-crete divergence operator when summed over the domain must reduce to a surfaceintegral at the domain boundaries.This is always satisfied when a finite-volume formof the divergence operator is used in the form of equation (42). However, deviationsfrom the effectively second-order divergence operator (high-order approximations)are difficult to reconcile and result in artificial mass sources/sinks in the compu-tational domain and incomplete convergence. It is noteworthy that equation (48)places no restriction on the discrete gradient operator. Following from equation(48) is the additional condition that results from Gauss divergence theorem appliedto the discrete pressure equation∫

S

(Gp) · ndS = 1t

∫S

u · ndS (49)

Hence, if the intermediate incompressible velocity field in the fractional step methodsatisfies global mass conservation, which it should, then the LHS of equation (49)should sum to zero. The use of a zero gradient condition on pressure at inhomoge-neous boundaries for known boundary velocities is consistent with this equation.The condition is also satisfied at homogeneous or periodic boundaries.

Under the requirements of eliminating grid scale oscillations on a nonstaggeredgrid and satisfying the integrability constraint and discrete continuity, one formu-lation that is quite effective is to interpolate the nodal Cartesian velocities to thecell faces (ui) to obtain the contravariant fluxes

√gU j. However, the interpolated

contravariant cell face fluxes are treated as velocities on a staggered grid and arecorrected by using the compact form of the pressure gradient. The net result of thisformulation is that the spurious pressure terms in equations (45) and (46) are elim-inated. However, the equivalent nodal momentum equation still carries the burdenof the spurious third derivative pressure term (see equation (41)). In spite of this,the problem of grid scale oscillations is considerably reduced and in sufficientlywell-resolved flows is non-existent. This is referred to as “current formulation” inthe following discussion.

The different formulations are tested in turbulent channel flow at Reτ = 180 ona 64 × 64 × 64 mesh with z+ = 8.8 and x+ = 35.2. The pressure spectrum is



1

0.01

Epp

Epp

0.001

10−5

0.0001

1 10 100Kz Kz

0.1

staggerednonstaggered (L22)nonstaggered (L23)nonstaggered (current)

1

10−8

10−7

10−6

10−5

0.01

0.1

0.001

1 10 100

staggered

y+ = 5 y+ = 120

nonstaggered (L22)nonstaggered (L23)nonstaggered (current)

Figure 6.5. One-dimensional pressure spectra in turbulent channel flow atReτ = 180 for different pressure formulations.

calculated near the wall at y+ = 5 and toward the center of the channel at y+ = 120and plotted in Figure 6.5. Different nonstaggered grid formulations are compared tothe staggered grid. L22 is the default noncompact form of the Laplacian constructedon a nonstaggered grid [14], L23 is the Laplacian constructed byTafti et al. [14], andlastly the current formulation is shown. The pressure spectrum at the two locationsshow that while there is some energy accumulation in the near-grid scales for all thenonstaggered grid formulations, the current formulation prevents a large buildup.Between the default formulation L22 and the designed formulation L23, L23 issomewhat better. At the same time, the presence of spurious oscillations near andat grid scales in pressure do not necessarily carry through to the power spectrumof the velocity components as seen in Figure 6.6. In fact it is only in the spanwisevelocity that any accumulation of energy is present.

Hence, the following series of steps are used in the corrector step. First, theintermediate cell face contravariant fluxes are constructed as follows:

√gU j = √

g(a j)iui (50)

by interpolating the nodal velocities at the cell face. The contravariant flux is thenconserved at nonmatching boundary faces to obtain 〈√gU j〉. Then, the correc-tion form of the nodal Cartesian velocities and cell face contravariant fluxes arewritten as:

un+1i = ui −t(a j)i

⟨∂Pn+1

∂ξj

⟩(51)

〈√g(U i)n+1〉 = 〈√gU i〉 −t√

ggik⟨∂Pn+1

∂ξk

⟩(52)



1

.01

0.01

0.001

0.0001

1 10 100

Euu

kz

y+ = 5

Staggerednon-staggered (L22)non-staggered (L23)non-staggered (current)

0.01

0.001

0.0001Evv

kz

y+ = 5

10−5

10−6

1 10 100


.01

0.01

0.001

0.0001

Ew

w

kz


y+ = 5

10−5

10−6

10−7

10−8

1 10 100

Figure 6.6. One-dimensional velocity spectra in turbulent channel flow atReτ = 180 for different pressure formulations.

Using the finite-volume divergence operator in the mass continuity equation tosatisfy the integrability constraint and substituting equation (52) for the cell facecontravariant fluxes, the pressure equation takes the form:

∂

∂ξj

(√gg jk

⟨∂Pn+1

∂ξk

⟩)= 1

t

∂〈√gU j〉∂ξj

(53)

The calculated pressure field at level n + 1 is then used to correct the nodalCartesian velocities and the cell face contravariant fluxes using equations (51)and (52), respectively. The use of 〈〉 on the pressure gradient term implies theintegral conservation of this quantity at nonmatching interface boundaries, dur-ing the solution of equation (53). Hence,

√g(U i)n+1 is automatically conserved

when the correction in equation (52) is applied to the conserved intermediatefield.



6.5.3 Integral adjustments at nonmatching boundaries

To eliminate the buildup of small interpolation errors, integral adjustments at non-matching interfaces are used for conserved variables. At the beginning of eachtime step, momentum and energy fluxes are conserved at nonmatching bound-aries by imposing the equality of mass or volume fluxes. With reference toFigure 6.2,

∑ξ+

〈√gU 1〉φ =∑η+

〈√gU 2〉φ (54)

in the ξ-η direction, where φ= ui or T are the interpolated values. The equality isused to obtain the conserved quantity <φ>.

The intermediate contravariant fluxes in equation (50) are first calculated fromthe interpolated intermediate velocities. The calculated fluxes, however, do notexactly satisfy the integral balance across the nonmatching interface. The conservedfluxes 〈√gU 1〉 are obtained by imposing the integral balance across the two facesas in ∑

ξ+

√gU 1 =

∑η+

√gU 2 (55)

There are two choices available in imposing equality of the conserved quan-tity in equation (54). One is proportional scaling and the other is an additiveadjustment. Proportional scaling may have difficulties when the mean mass fluxacross the boundary is close to zero, whereas additive adjustments may smearweak recirculation zones at or near the interface. Since most of the time ina well-resolved flow, the interpolation error is small, the contravariant flux isadjusted linearly for the sake of robustness. However, for small values of

√gU1

in equation (54), even an additive adjustment has the potential of leading toinstabilities, since to obtain <φ> the total flux needs to be divided by thisquantity.

For the pressure equation (53), the pressure at the ghost cell is first inter-polated from the relevant boundary that serves as a boundary condition for theinterior values. Since in incompressible flows the pressure gradient is the relevantquantity that drives the flow, this quantity is conserved at nonmatching interfacialboundaries. For each iteration in the pressure equation solver, the normal gradi-ent of pressure is first evaluated and then integrated over the relevant interfacialboundary. Based on the difference of the two integral values, a constant valueis then added/subtracted to the pressure at the ghost cells in order to guaranteethe equality of the two integral values. Since this procedure is integrated into thelinear solution of the pressure equation, it automatically guarantees an integral bal-ance of the contravariant fluxes across nonmatching interfaces calculated throughequation (52).



6.6 Discretization of Convection Terms

By far, from the numerical standpoint, the discretization of the nonlinear convectiveterms has the most profound effect on the accuracy with which turbulence is rep-resented on a given mesh. An important aspect of turbulent flow simulations is therole of truncation and aliasing errors, which usually manifest themselves in the highwavenumbers of the resolved scales. Ideally, the discretization should have the capa-bility of representing all the grid-resolved wavenumbers with accuracy, conservequadratic quantities such as kinetic energy, have minimal aliasing errors, and bestable and robust at the same time. While truncation errors can be directly estimatedfromTaylor series expansions in physical space and modified wavenumber represen-tations in wave space (Lele [17], Beudan and Moin [18]), the quantification of alias-ing errors in time integration algorithms and their effect on the accuracy and numer-ical stability of the algorithm is not straightforward (Kravchenko and Moin [19]).

The most popular method for approximating convection terms has been an SOCdifference discretization on staggered as well as nonstaggered grids. Although notperfect, it provides a reasonable and economical alternative to other high-orderschemes. When the SOC scheme is applied to the divergence or conservative formof the convective terms in a finite-volume framework, it has the desirable propertyof conserving global kinetic energy. Although total kinetic energy is conserved,also present is the artificial redistribution of energy among the resolved scales.Related to this are aliasing errors that result from the spurious accumulation of unre-solved numerical subgrid energy in the resolved scales due to the high-wavenumbernonlinear interactions of the velocity field. Early work in the meteorological andgeophysical community (Phillips [20], Lilly [21], and Arakawa [22]) has addressedthe role of aliasing in the development of nonlinear numerical instabilities, which ledto the development of global energy-conserving finite-difference schemes. Globalenergy-conserving schemes require that the advection of a constant velocity fieldshould not produce nor destroy global kinetic energy. According to Arakawa [22],the two conditions that a convective differencing scheme should satisfy to conservequadratic quantities are:

Ai′′, j′′, k ′′ : i, j, k = −Ai, j, k : i′′, j′′, k ′′Ai, j, k : i, j, k = 0

(56)

The first equation requires that the discrete effect of one node on another shouldbe equal and opposite, and the second condition requires that the node should notinfluence its own value. In 1-D, AE(i) = −AW(i + 1) and AP = 0, where E, W, and Prefer to a three point finite-difference stencil. For the finite-volume approximations,the first condition is always satisfied, whereas the second condition is linked to thesatisfaction of discrete continuity in a form that will lead to it being zero. Hence,the energy-conserving property is inexplicably linked to the consistent solutionof the pressure equation. It can be easily shown that the SOC scheme used witha consistent second-order construction of the divergence operator will satisfy thesecond condition. Any deviation from SOC to high-order approximations requiresthat the divergence operator in continuity and pressure be tailored to satisfy the



second condition. In fact most other conventional high-order schemes based onupwinding or central-differencing do not satisfy the second condition and specialconsiderations have to be given to construct high-order schemes that are energyconserving (Morinishi et al. [23]).

On the other hand, in spite of not conserving global energy, high-order finite-difference schemes have been formulated and used to reduce the truncation errorsassociated with the finite-difference approximations. A number of such formula-tions have been used in the past based on high-order compact differences (Lele [17]),conventional high-order upwind biased approximations (Rai and Moin [24]), andfourth-order central differences. Accompanying the high-order convective schemesare also high-order approximations of the viscous terms and the pressure equation.Tafti et al. [25] have investigated a number of high-order schemes based on con-servative and nonconservative formulations of the convection term combined withdifferent pressure equation formulations.

A few of the high-order formulations were tested for the numerical simulationsof turbulent channel flow at Reτ = 180 (based on wall friction velocity and channelhalf-width) using a staggered grid. Results from the 2c22, 5c22, 5nc22, and 5nc44formulation are compared with the spectral simulations of KMM [6] and experi-ments of Kreplin and Eckelmann [26] (KE). Here c stands for the conservative ordivergence form of the convection terms, which is discretized in a finite-volumeframework, and nc stands for a nonconservative form of the convection term, whichis discretized using finite-difference approximations. The first number denotes theorder of approximation of the convection term, 2 stands for SOC, and 5 for fifth-order upwind biased approximations. The last two numbers denote the order ofapproximation of the divergence and gradient operator used to construct the pres-sure equation Laplacian: 22 refers to SOC approximations whereas 44 denotesfourth-order central approximations. The high-order methods are accompanied bysixth-order central interpolation operators and sixth-order central treatment of thediffusion terms. The 5nc44 formulation does not satisfy the integrability constraintin the inhomogeneous wall normal direction. Only the 2c22 scheme is globallyenergy conserving.

The channel computational domain extends from 0 to 4π in x, −1 to 1 in y,and 0 to 2π in z, and calculations are performed with grids of 64 × 64 × 64 and128×96×128.A semi-implicit formulation is used to obtain the intermediate veloc-ity field. The pressure equation is solved by using 2-D FFTs in the homogeneousx- and z-directions with a line inversion in the y or inhomogeneous direction for eachcombination of wavenumbers. A nondimensional time step oft = 2.5 × 10−3 and1.0 × 10−3 is used for the coarse and fine grid simulations, respectively. Velocityfields from previous simulations are used as initial conditions and each case is runfor 10 nondimensional time units before any statistics are collected. The statis-tics are collected for additional 10 nondimensional time units. Figures 6.7 and 6.8show some representative results for the coarse grid and fine grid simulations,respectively. The predicted mean streamwise velocity profiles in wall coordinatesare compared to the theoretical laminar sublayer profile (u+ = y+) and the log-law profile of u+ = 2.5ln(y+) + 5.5, together with predictions of kinetic energy



20

(a) (b)

2c22

5c22

5nc225nc44u⊥ = yu⊥=2.5In(y⊥)+5.5u+

y+

15

10

5

00.1 1 10 100

5

4

3

2

1

0

k

y +0 20 40 60 80 100

2c22

5c225nc22

5nc44KMM

1

0.5 +

++ + +

(c) (d)

0

−0.5

−1

−1.5−2.6

s(v)

y +

0 20 40 60 80 100

2c225c225nc225nc44

KMM

Euu

Kx

1

0.1

0.01

0.001

0.0001

10−5

10−6

10−7

10−8

1 10 100

2c225c225nc225nc44

KMM

Figure 6.7. Comparison of different convection and pressure formulations in turbu-lent channel flow at Reτ = 180 at a resolution of 643. (a) Mean velocityin wall coordinates; (b) turbulent kinetic energy distribution; (c) one-dimensional streamwise velocity power spectrum in z-direction; and(d) skewness of cross-stream velocity fluctuations.

distribution in wall coordinates, 1-D energy spectrum of the streamwise velocity(Euu) at y+ = 5.6, and third-order moment or skewness of the cross-stream velocity,which is very sensitive to the numerical approximation. The following observationscan be made:

1. In both the coarse grid and the fine grid simulation the 2c22 formulation com-putes the spectrum Euu with the best accuracy.The high-order schemes 5c and 5nc

both exhibit a dissipative behavior in representing the spectrum. The dissipativenature of the 5c and 5nc formulations is also evidenced by the overprediction ofthe core velocity and the overprediction and right-shift of the maximum turbulentkinetic energy profiles for the coarse grid simulations.



5

3

2

1

00 20 40 60 80 100

0.0001

0.001

0.01

0.1

1

10−8

10−7

10−6

10−5

2c225c225nc225nc44KMM

1 10

(a) (b)

100

k Euu

kxy+

4

2c225c225nc225nc44KMM

Figure 6.8. Comparison of different convection and pressure formulations in tur-bulent channel flow at Reτ = 180 at a resolution of 128 × 96 × 128.(a) Turbulent kinetic energy distribution and (b) one-dimensionalstreamwise velocity power spectrum in z-direction.

2. There are very small differences observed in the prediction capability of 5nc44and 5nc22, in spite of the former not satisfying the integrability constraint (equa-tions (48) and (49)).All of them predict the third moment or skewness with muchbetter fidelity than the second-order scheme.

3. As the grid resolution increases to 128 × 96 × 128, the high-order upwindschemes exhibit better accuracy in predicting the turbulent kinetic energy dis-tribution compared to the 2c22 formulation. However, the 2c22 formulation isbetter at predicting the spectral energy distribution.

It can be surmised from these results that high-order upwind schemes in spite oftheir dissipative nature are still very usable and at high resolution in the DNS limitgive more accurate results of turbulent statistics than the SOC difference scheme.However, they are computationally expensive because of the larger directionallydependent finite-difference stencils.

In complex CFD problems with complex grids, there is often the temptation touse low-order schemes such as second-order upwinding or the third-order upwindbiased approximation (QUICK) or other monotonicity preserving schemes becauseof their relative stability and lower computational cost. While these schemes areuseful in steady RANS calculations, they can lead to erroneous results in LES andDES. To elucidate on this, a normalized variable diagram (NVD) (Leonard [27]) isused to construct the total variation diminishing (TVD) schemes below. The NVDis constructed by defining the upstream and downstream directions based on thecell face flux direction as in Figure 6.9.

Then the normalized variable at any point is defined as:

φ = φ − φu

φd − φu



CV CV

φd

φd

φf φf

ufuf

φcφcφu

φu

Figure 6.9. Nomenclature used to construct the NVD.

−0.5−1

−0.5

0

0

0.5

1

1.5

2

0.5 1 1.5

φc∼

φf = φc∼ ∼

φf = 0.5(1+φc)∼ ∼

φf = 2φc∼ ∼

∼φfSecond-order centraldifference with TVD limiter

−0.5−1

−0.5

0

0

0.5

1

1.5

2

0.5 1 1.5

φc∼

φf = φc∼ ∼

φf = 3/8 + 3/4φc)∼ ∼

φf = 2φc∼ ∼

∼φf

∼φf = 1

Third order upwind-biasedwith TVD limiter

Figure 6.10. NVD diagram for TVD-limited second-order central scheme and theQUICK scheme.

where ∼ denotes the normalized variable. For example, an SOC difference approx-imation at the cell face at time step ntφn

f = (φnc + φn

d )/2 is written in NVD

coordinates as φnf = (1 + φn

C )/2. The equivalent TVD limiter in NVD coordinatesis given by:

φnf = φn

c for φnc ≤ 0 and φn

c ≥ 1

φnC ≤ φn

f ≤ min [2φnC , 1] for 0 ≤ φn

c ≤ 1(57)

Effectively, the first condition uses a first-order upwind approximation if φnC is not

bounded by φnu and φn

d . Within the bounds established in the first condition, thesecond condition further limits the face value to lie between between φn

C and theminimum of φn

d and φnc + (φn

c − φnu). The limiter applied together with the SOC

scheme is illustrated in Figure 6.10 on the NVD. Similarly, the QUICK scheme innormalized coordinates is expressed as φn

f = 3/8 + 3/4φnC .

In contrast to the TVD limiter, which is applied to the convected variable at ntand hence can be too restrictive, Thuburn [28] developed a multidimensional fluxlimiter based on the less restrictive universal limiter (UL) proposed by Leonard [27].



0.10

5

10

u+

15

theoretical

SOC

SOC + TVD

SOC + DSM

SOC + UL

QUICK

20

25

1

(a) (b)

(c)

100 00

1

2

3

4

5

6

7

50 100 150 200y+

kz

y+

k

DNS (KMN)

SOC

SOC + TVD

SOC + UL

SOC + DSM

QUICK

10−6

1 10

Euu

100

10−5

0.0001

0.001

0.01

0.1

1

SOC

DMS(KMM)

SOC + TVD

SOC + DSM

SOC + UL

QUICK

Figure 6.11. Prediction capability of different convective schemes in turbu-lent channel flow at Reτ = 180. (a) Mean velocity profile inwall coordinates; (b) turbulent kinetic energy distribution; and(c) one-dimensional energy spectrum of streamwise velocity in thez-direction.

In this scheme, the intermediate velocities are first calculated with the base approx-imations and then checked for monotonicity in a multidimensional framework.By not applying the monotonicity preserving criteria at time level n, the univer-sal limiter is much less restrictive than the TVD limiter. The effect of differentschemes is tested in turbulent channel flow at Reτ = 180 with streamwise grid spac-ing x+ = 35.34, spanwise z+ = 8.84, and cross-stream 0.63 ≤y+ ≤ 8.11 fora domain of 4π × π × 2 in x, z and y, respectively and plotted in Figure 6.11.The results are compared with DNS data of KMM. In fully developed channel flowsince the flow velocity adjusts to the constant pressure gradient in reaction to thesimulated wall friction, core velocities larger than theory are indicative of a dissi-pative scheme. This is also typically accompanied by high turbulent kinetic energywith the maximum shifted toward the core of the channel. The energy spectrumalso provides additional evidence of the spectral properties of the scheme. Based



on these criteria, out of all the schemes the TVD limiter is the most dissipativeeven when used with the nondissipative SOC scheme. The QUICK scheme is notas dissipative as the TVD limiter, but in spite of having a higher accuracy thanthe SOC scheme it fails to give accurate results. On the other hand, the universallimiter behaves very much like the dynamic Smagorinsky model [29] (DSM) andseems to be a viable alternative for turbulent flow simulations. The use of minimalnumerical dissipation as in the SOC-UL method to substitute for explicit subgridmodeling as a means for performing LES is referred to as implicit LES or ILES inthe literature (Margolin et al. [30]; Grinstein et al. [31]).

In summary, discrete approximations used for the nonlinear convection termshave a large bearing on the accurate prediction of turbulence in time-dependentmethods based on DNS, LES, and DES. The SOC method used in its conservativeform is a good compromise for complex geometries between overly dissipative low-order (second- and third-order upwind scheme) and the extra cost and complexityof high-order methods.

6.7 Large-Eddy Simulations and Subgrid Modeling

Central to the large-eddy simulation of turbulence is the filtering operation andmodeling of the resulting subgrid stresses. In generic terms the filtering operationis defined by a low-pass 3-D filter in physical space:

f (xi, t) =+∞∫

−∞

3∏i=1

Gi(xi − x′i)f (x′

i, t)dx′i (58)

In any discrete simulation by definition there is an implicit filter inherently appliedto the calculation. In physical space with finite-difference/finite-element methodsthe implicit filter is referred to as the top hat filter because of its shape and isgiven by:

Gi(xi − x′i) = 1

xi; |xi − x′

i| ≤ xi/2

= 0; otherwise(59)

where xi is the grid spacing. Similarly in spectral simulations the filter width inwavenumber space is implicitly decided by the number of fourier modes used inthe calculation and is referred to as a sharp cut-off filter and is given by:

Gi(ki) = 1; ki ≤ kc

= 0; otherwise(60)

where kc is the cut-off wavenumber. In contrast, the top hat filter is not sharp inwavenumber space but smooth such that it partially filters even the high wavenum-bers resolved on the grid (Sagaut and Meneveau [32]). Another smooth filter thathas found some use for explicit filtering is the Gaussian filter. There has been muchdebate and research on the properties of these filters and whether explicit filtering



with a filter width larger than that of the implicit grid spacing should be used infinite-difference/finite-volume methods. Since finite-difference methods inevitablyresult in some corruption of the high wavenumbers of the resolved field which arealso used to model the subgrid stresses particularly in models that assume a simi-larity between the high wavenumber resolved field and the subgrid field, numericalinaccuracies and modeling errors become inextricably linked together. Hence theargument for explicit filtering is to break this link by using a twice as fine grid andthen explicitly filtering out the numerically corrupted scales and using only the 2filtered scales to model the subgrid stresses. However, the cost of the simulationmore than doubles (in 1-D) and studies have shown that calculations on the twice asfine grid without explicit 2 filtering give better predictions of turbulent quantities.

In the past, efforts on subgrid modeling focused on calculating the Leonardterms and on modeling the individual terms in equation (21) (Leonard [33]; Clarket al. [34]). However this effort was largely abandoned in favor of modeling allthe subgrid terms together. There are a number of subgrid-scale models varyingin complexity from eddy viscosity to one equation models. Reviews can be foundin Rogallo and Moin [35], Ferziger [36], Lesieur and Metais [37], and in the textby Saugat and Meneveau [32]. The most widely used closure model, suggestedby Smagorinsky [38], is based on the eddy-viscosity approximation in which thesubgrid-scale stresses are related to the strain rate tensor of the resolved field throughan eddy viscosity (equation (22)). The eddy viscosity is computed from the resolvedstrain rate magnitude and a characteristic length scale (equation 61). The lengthscale is assumed to be proportional to the filter width via a Smagorinsky constant,which has assumed values anywhere from Cs = 0.06 to 0.23 in past studies.

1Ret

= C2s (

√g)2/3|S| (61)

yielding

τaij = −2C2

s (√

g)2/3|S|Sij = −2Cβij (62)

where C = C2s , |S| =

√2SijSij is the magnitude of the strain rate tensor that pro-

vides the nondimensional time scale of subgrid turbulence. Germano et al. [29]proposed a dynamic procedure for the computation of the Smagorinsky constant.The dynamic procedure utilizes information from the resolved high wavenumberscales by applying a test filter in addition to the grid filter to obtain the modelconstant.

Central to the dynamic procedure is the application of a test filter denoted by Gto the filtered governing equations (10) and (11) with the characteristic length scaleof G being larger than that of the grid filter, G (usually = 2). In computationalcoordinates the filter is written as:

G(ξi − ξ′i) = 1

2ξi; |ξi − ξ′i | ≤ ξi

= 0; otherwise; i = 1, 2, 3 (63)



and the filtered quantity as:

f (ξi) =+ξI∫

−ξi

3∏i=1

1

2ξif (ξ′i′ ) dξ′i (64)

Using Simpson’s rule for a 1-D integration (say; i = 2 or η-direction)

f (η) = 1

6( f j−1 + 2f j + f j+1) (65)

Using a five-point integration rule

f (η) = 1

2

(1

6( f j−1 + 4f j−1/2 + 2f j + 4f j−1/2 + f j+1)

)(66)

Using a second-order interpolation at the cell faces in equation (66) leads to thetrapezoidal top hat test filter

f (η) = 1

4( f j+1 + 2f j + f j−1) (67)

From the development above it is straightforward to extend the filtering operationto nonuniform grids and to inhomogeneous boundaries. For example, at a wallboundary a simple extension of the method above will give:

f 2(η) = 14

(2f 1 + f 2 + f 3) (68)

where 1 denotes the boundary node and node 2 is at /2 from the boundary. Theone-sided filter now has a complex transfer function and introduces phase errors inthe filtering operation. Najjar and Tafti [39] studied the effect of inhomogeneousfiltering in turbulent channel flow and found that the local modification to the testfilter had an effect on the distribution of the dynamic constant and subsequently theturbulent viscosity at the first few nodes near the boundary.

Using the same methodology a number of “high-order” filters can be constructed(Najjar and Tafti [40]). The high-order filters filter out a smaller percentage of thehigh wavenumber resolved scales and approach the behavior of a cut-off filter. Itwas determined by Najjar and Tafti [41], that the trapezoidal filter yielded moreconsistent results with finite-difference approximations.

Since the grid filter is applied to the conservation equations in physical space,for consistency the test filtering operations are transformed from computational to



physical space. This is accomplished by the following transformations:

f x = 1

x

[xξ f (ξ) + xη f (η) + xζ f (ζ)

](69)

f xy = 1

y

[yξ f x(ξ) + yη f x(η) + yζ f x(ζ)

](70)

f xyz = 1

z

[zξ f xy(ξ) + zη f xy(η) + zζ f xy(ζ)

](71)

to give the filtered variable in physical space. Contrary to the formulation of thesubgrid terms and consequent test filtering in physical space, Jordan [42] formulatedthe subgrid stresses in computational space, i.e., the grid filtering operation isapplied to the transformed equations.

The test filtered subtest scales that include part of the grid-resolved scales andthe subgrid scales Tij = uiuj − uiuj are modeled as:

T aij = −2C2

s (σ2(√

2)2/3)|s|sij = −2Cαij (72)

where σ is the ratio of the test filter width to the grid filter width. The most commonvalue used in the literature is 2, which is consistent with a sharp cut-off filter. Forthe trapezoidal filter, Najjar and Tafti [41] obtained a value of

√6

The resolved turbulent stresses representing the energy scales between the testand the grid filters, Lij , are then calculated and related to the subtest (Tij) and subgrid(τij) stresses through the identity

Lij = Tij − τij, where Lij = uiuj − uiuj (73)

to give

Laij = Lij − 1

3σij Lkk = −2Cαij + 2Cβij (74)

Equation (74) corresponds to a set of five independent equations that cannot besolved explicitly for C, which appears in the test filtering operation.A simplificationarises if C is assumed only to be a function of time and the direction where noexplicit test filtering is applied (say the y-direction), i.e., C = C(y, t), then C canbe extracted from the filtering operation resulting in Cβij = Cβij. Using a least-square minimization procedure (Lilly [43]) for flows with at least one homogeneousdirection, the following closed form for the dynamic subgrid-scale model constant,C, is obtained:

C(y, t) = −1

2

〈Laij(αij − βij)〉

2〈(αmn − βmn)(αmn − βmn)〉(75)

where 〈〉 represents an averaging operation in the homogeneous directions. Equation(75) has been extensively used in LES computations of various turbulent flows.



The averaging procedure applied in equation (75) represents a smoothing operatorto damp the large local fluctuations in C. In complex geometries however, there areno homogeneous directions and the test filter is often applied in all three directionsand C has to be computed locally. The simplest approximation is to remove Cfrom the filtering operation in equation (74) and then apply some smoothing to thelocal values of C based on zonal averaging or application of a smoothing filter notvery different from the test filter. In addition to the smoothing, some bounds areusually placed on C or 1/Ret . The dynamic procedure is capable of predictingC< 0 or backscatter of energy. To permit backscatter, but to prevent negativeeffective viscosities and the ensuing numerical instability, −1/Ret is limited by1/Re. However, usage in complex geometries with finite-difference approximationshas tended to eliminate backscatter by constraining C ≥ 0. Just as negative valuesof 1/Reeff can induce numerical instabilities, so can very high positive values of Cby decreasing the effective Reynolds number of the flow, which could reduce theeffective diffusive time scale below the time step. Hence a somewhat arbitrary upperbound on C of the order 0.04 or Cs ≤ 0.2 is also used based on the theoreticallyderived value of Cs = 0.18 for homogeneous turbulence. More elaborate solutionsfor calculating C from equation (74) can be found in Ghosal et al. [44] and Meneveauet al. [45].

Table 6.1 summarizes calculations done in a fully developed ribbed duct fromTafti [46] at a nominal Re = 20,000. The duct is of square cross section with asquare rib of dimension e/Dh = 0.1 and a rib pitch of P/e = 10, where e is thedimension of the rib and P is the streamwise pitch between ribs. Four calculationsare shown at different resolutions to highlight the effect of the dynamic Smagorinskymodel (DSM in Table 6.1). An SOC difference scheme is used for the convectiveterms. The dynamic Smagorinsky constant is localized and is constrained to bepositive. The turbulent Prandtl number was fixed at 0.5 [47]. The heat transferresults were found to be quite insensitive to the turbulent Prandtl number and testson a grid of 1283 showed no difference when Prt was changed to 0.9. Two gridsizes of 963 and 1283 are investigated with and without the model. Since Nusseltnumber augmentation ratio is a very weak function of Reynolds number in the range20,000 to 30,000, the predictions are compared to the available experiments of Rauet al. [48] at Re = 30,000. The table summarizes the prediction of reattachmentlength of the separation bubble behind the rib, and Nusselt and friction coefficientaugmentation.

All calculations reproduce the major flow structures with fidelity, namely theeddy formed at the junction between the rib and the wall, a recirculation zoneformed on top of the rib and behind the rib with a corner eddy, and the lateralimpingement of flow on the smooth side wall at the rib junction. Qualitatively, thebulk flow field results are indistinguishable, but quantitative differences of 10 to15% exist between the different calculations. However, there are large differencesin the predicted heat transfer and friction coefficients. The degree of underpredic-tion of heat transfer and friction varies from 20 to 30% for the 963 quasi-DNScalculation to 15 to 20% for the 1283 quasi-DNS. This is caused primarily by lowturbulence intensities. The use of LES with the dynamic model increases the level



Table 6.1. Summary of heat transfer and friction data and percentage error withdata of Rau et al. [48]. Experimental uncertainty is ±5%

Computations Rau et al. [48],e/Dh = 0.1, P/e = 10 e/Dh = 0.1,

P/e = 10

963 963 DSM 1283 1283 DSM

Reτ 6,667 6,667 6,667 6,667 –

Reb 24,000 20,000 22,000 20,000 30,000

% Form loss 92 94 92 91 85

Reattachment 4.8 4.3 4.6 4.1 4.0–4.25length (xr/e)

<Nu>/Nu0 (Nu0 = 0.023 · Re0.8b · Pr0.4)

Rib 2.22 2.84 2.54 2.89 –

Ribbed wall 1.78 2.35 2.00 2.4 2.40(−26%) (0%) (−17%) (0%)

Smooth sidewall 1.40 1.89 1.60 1.89 2.05(−32%) (−7%) (−22%) (−7%)

Overall with rib 1.67 2.22 1.89 2.23 –

Overall w/o rib 1.58 2.11 1.79 2.14 2.21(−28%) (−4.5%) (−19%) (−3.4%)

f /fo(f0 = 0.046 · Re−0.2b )

Overall 6.11 8.53 7.23 8.6 9.5(−36%) (−10%) (−24%) (−9%)

of turbulence in the flow, particularly near walls, and is able to predict the heattransfer coefficient to within experimental uncertainty and friction coefficient towithin 10% of experiments for both mesh resolutions. The level of turbulenceaugmentation provided by the dynamic model is commensurate with the mesh res-olution such that the turbulent energy, heat transfer coefficient, and friction arepredicted at the right levels independent of the resolution.

The dynamic Smagorinsky model has been used extensively in the literaturefor complex flows, and in spite of some as yet unresolved issues it does a reason-able job in calculating the eddy viscosity in complex flows. Chief among them isthat it is difficult to separate out numerical errors from the subgrid model. Somenew developments in subgrid modeling and LES simulations can be found in thedefiltering approach to reconstruct the subgrid scales (Domaradzki and Saiki [49];Domaradzki and Loh [50]; Stolz and Adams [51]), the use of grid hierarchies fordynamic simulations on coarse grids using information from fine grids to model



the subgrid stresses with large savings in computational effort (Dubois et al. [52,53] , Terracol et al. [54]), and the variational multiscale method of Hughes [55, 56].Alternate methods of simulating turbulent flows without explicit subgrid modelinghave also been proposed and used [31].

6.8 Detached Eddy Simulations or Hybrid RANS-LES

LES is an attractive alternative to DNS, but is still computationally expensive inwall-bounded flows, particularly when wall friction and heat transfer coefficientsare sought from the calculation. Typically, to capture the energy-producing scalesa fine grid resolution is required not only in the wall normal direction (which evenwall integration or low Reynolds number RANS models require), but also in thewall parallel direction. In typical turbulent boundary layers the streaks and bursts,which are primarily responsible for near-wall production of turbulence, extendabout 40 spanwise wall units and are spaced about 100 wall units apart. With LESit is imperative that these structures are resolved in the calculation, which puts largedemands on the near-wall resolution as the Reynolds number increases. Hence it isnot surprising that attempts have been made to use wall functions with LES (Cabotand Moin [57]; Piomelli and Balaras [58]).

The forbidding computational expense of LES at high Reynolds coupled with theinability of RANS models to accurately predict separation and reattachment led tothe development of DES, or more generally a hybrid RANS-LES method, by Spalartet al. [1]. The motivation behind the formulation was to allow a RANS solution inthe near-wall region which would default to LES away from the wall. Hence, insome sense it also acted as a wall model, wherein by using RANS very close to thewall, the strict resolution requirements of LES in the wall parallel direction could berelaxed considerably.The original formulation was introduced within the frameworkof the Spalart–Almaras [59] (SA) model. What differentiated the DES method fromprevious efforts to combine URANS and LES was the fact that a single model (inthis case the SA model) acted both as the RANS model in the near-wall region andas an eddy-viscosity subgrid model in regions away from the wall. The switch in theintended use of the model occurs transparently and does not involve complicatedmatching of flow and turbulence quantities at the interface. In the case of the SAmodel, this is incorporated by comparing the distance to the wall (d) with the localcell size () (which in LES is representative of the smallest resolved scale). Ford <Cdes, the SA model is used in the RANS mode and when d ≥ Cdes, Cdes isused as the characteristic turbulence length scale for operation in LES mode. Thisidea was applied successfully to airfoils with massive unsteady separation (Shur etal. [60]), to external flow over cylinders (Travin et al. [61]), and to turbulent channelflow at high Reynolds numbers (Nikitin et al. [62]) and to a number of externalflows with separation [63–66]. In initiating the switch from RANS to LES, whilethe formulation in the SA model related the turbulent length scale to the distancefrom the wall, Strelets [67] extended the DES concept to a two equation eddy-viscosity model (SST model) by calculating the turbulent length scale from the



DES

1.000.800.600.400.200.00

Rib

Figure 6.12. Plot of the time-averaged LES and RANS region in the DES com-putation for a 643 grid in a lateral plane. A value of 1 represents acomplete LES region and a value of 0 a complete RANS region. Flowis periodic or fully developed in the streamwise direction. Reynoldsnumber based on mean flow velocity and hydraulic diameter of ductis 20,000 [69].

model turbulent quantities. This is illustrated for the k −ω model (Wilcox [68]) inwhich the dissipation of turbulent energy, −Cµkω is written as k3/2/ lDES. The DESlength scale is obtained from lDES = min(lRANS,lLES), where lRANS = √

k/(Cµω)and lLES = CDES max(x,y,z) is the turbulent length scale and the grid lengthscale, respectively. For a grid scale smaller than the turbulent length scale the modelreverts to an LES solution. This feature facilitates the computation to be cognizantof the eddy length scales and hence behave as RANS or LES depending on theinstantaneous local conditions, unlike in the case of SA model DES, where it isconstrained to be a wall model. Another prominent feature of the DES version ofthe two equation model is that although an instantaneous discontinuity may existbetween the RANS region and the LES region, in the mean however a smoothtransition takes place from RANS to LES and vice versa. Figure 6.12 plots thefraction of time the model operates in LES mode at a given location in space. It canbe observed from Figure 6.12 that the RANS regions transition into LES smoothly.In a DES SA model, the distance from the wall determines the switch from RANSto LES, which is not as receptive to instantaneous flow features.

Much like the SA model the near-wall region is always resolved in the RANSmode, which transitions to the LES mode as the distance from the wall increases.The flow in the vicinity of the ribs is mostly resolved in LES mode all the wayto the channel center. This includes the unsteady large-scale dynamics of theseparated shear layer as shown in Figure 6.12. Though the inter-rib spacing ispredominantly LES, it transitions to a RANS mode at the center of inter-rib region.The grid facilitates the computation of the region in RANS and/or LES dependingon the instantaneous conditions existing in the region. Downstream of the rib, inthe region of separation it is observed that the computation is mostly carried outin the LES model. This sensitizes the RANS model to grid length scales, thereby



allowing the natural instabilities of the flow in this region to develop. Thus, theunsteady physics in the separated region is captured accurately.

In spite of its attractiveness and successful application, extension of DES to awide range of flows poses several challenges [69]. DES is a strategy that dependson the grid not only for accurately resolving the turbulence in the regions of interest,but also in determining the switch between RANS and LES.Though guidelines havebeen prescribed for SA-based DES (Spalart [70]) in external flows, grid generationin internal flows is not trivial and requires some a priori knowledge of the flow. Oneprobable alternative is the scale adaptive simulation (SAS) approach suggested byMenter et al. [71, 72] that compares a von Karman length scale to the turbulentlength scales to make the RANS-LES switch independent of the grid.

DES still depends on the prediction capability of the base RANS model topredict phenomena such as separation, transition, and effects of secondary strainrates due to rotation and buoyancy forces. In previous investigations it has beenfound that while DES was successful in capturing secondary strain effects, whichthe base RANS model did not capture when used in RANS or URANS mode ([73,74]), it also failed to predict shear layer transition in contrast to LES which wassuccessful [75].

The transition from URANS to LES still remains a “grey area” in DES. Thephysics behind the switch from a completely modeled solution in the RANS regionsto a well-resolved LES still remains to be understood [76].

6.9 Solution of Linear Systems

Solution of linear systems generated in the semi-implicit and implicit treatmentof the momentum equations and the pressure equation requires careful consid-eration because these take up a majority of the computational time. The systemmatrix in generalized coordinates with second-order approximations is sparse with18 off-diagonal bands on nonorthogonal grids and can either be symmetric ornonsymmetric. A symmetric system is only generated on orthogonal grids in thesemi-implicit treatment of momentum and in the pressure equation, whereas theimplicit treatment of convection always leads to a nonsymmetric system. Non-conformal block boundaries can further lead to strong asymmetries in the systemmatrix. There are a variety of methods for solving large sparse systems of linearequations (Meurant [77]). Out of these methods, iterative Krylov subspace meth-ods have proven to be quite efficient for CFD applications (Barrett et al. [78]). Whilethe method of conjugate gradients (CG) is useful for symmetric systems, methodslike BiCGSTAB and GMRES(m) can be applied to nonsymmetric systems (Tafti[79]). Preconditioning can be applied using a variety of methods. One particularframework that is quite effective is based on a two-level additive/multiplicativeSchwarz domain decomposition (DD) method (Smith et al. [80]; Dryja andWidlund [81]). In this method, the full system of equations is substructured intosmaller sets of overlapping domains, which are then solved individually in an iter-ative manner, all the time updating the boundaries such that the global system is



1 100

500

1000

MF

LOP

/S

1500

2000

100 1000Block size

104 105 106 107

Intel lll (1 GHz)Itanium l (800 MHz)IBM Power4 (2.3 GHz)Intel Xeon (3.06 GHz)Intel Itanium 2 (1.3 GHz)Apple G5 (2.3 GHz)

Figure 6.13. Effect of substructured block size on single-processor performanceon some modern computer chips. MLOP, million floating pointoperations.

driven to convergence. Each subdomain is smoothed with an iterative method suchas the Jacobi method or symmetric successive over-relaxation (SSOR) or incompleteLU (ILU) decomposition.

Not only does the DD method act as an effective preconditioner for the Krylovmethod, but also results in large performance gains on modern computer architec-tures. By substructuring the large system into smaller systems that are designed tofit into cache memory, main memory accesses are minimized, which result in largeperformance gains [82–84]. Figure 6.13 shows the effect of the sub-structured blocksize on performance with an additive Schwarz preconditioned conjugate gradient(ASPCG) algorithm for solving a linear system generated by a Laplace equation inCartesian coordinates.

Figure 6.13 shows that there is an optimal block size that depends on the pro-cessor speed, cache size, speed and memory bandwidth, and latency. For verysmall blocks the cache lines are not utilized effectively and the overhead of gettingdata from memory exceeds the equivalent number of floating point operations per-formed on the data. As the block size increases, the overhead of memory accessesis amortized over more floating point operations on block data in cache, and henceperformance increases till the block size gets so large that data required by theiterative solver for a given block does not fit in cache and has to be accessed frommain memory multiple times – once again increasing the overhead of data reads andstores and decreasing performance. The asymptotic performance beyond a block



size of 105 is an indicator of the main memory performance for each architecture.Figure 6.13 shows one aspect of the equation that governs time to solution. Theother effect that needs to be considered in designing an optimal block size is itseffect on number of iterations to convergence. Typically as the block size decreases,convergence slows and the number of iterations increase. This is countered by usinga two-level method in which a coarse level is implemented with a granularity ofone coarse node per substructured block. A typical V-cycle between the two levelsworks much like in multigrid, in which the solution is relaxed individually in eachblock, then restricting the block-integrated residues on to the coarse level followedby an iterative smoothing of a correction, and finally a prolongation of the correc-tion back to the blocks to be distributed across all nodes contained in the block.The additive Schwarz preconditioner provided the following benefits (Wang andTafti [82]):

• Reduced the number of iterations for convergence in the Krylov solver by a factorof 2 to 3, hence reducing the number of global inner products and matrix-vectorproducts.

• Increased floating point performance by a factor of 2 to 3.• Decreased the overall CPU time by nearly an order of magnitude.

6.10 Parallelization Strategies

It would not be an overstatement to say that much of the gains in the widespreaduse of steady and time-dependent CFD has been facilitated by the exponentialincrease in computational power over the past two decades. Not only have pro-cessors become more powerful, but the ability to harness their collective power inparallel has also increased. While single-processor performance is dependent onthe efficient use of fast cache, parallel performance is dependent on how efficientlya code parallelizes on a given architecture. Three distinct brands of parallel archi-tectures have emerged – shared, distributed, and distributed-shared memory (DSM)architectures. In the shared memory architecture, multiple processors have accessto the same shared memory through cache-coherent nonuniform memory accesses(cc-NUMA), whereas in the distributed memory architecture, the processors aredistributed across a network with each processor only having access to its ownmemory. Hence, explicit messages are required to transfer data from one mem-ory to the other. The DSM architecture combines the two architectures by havingmultiple shared memory nodes distributed across a network.

Barring recent research efforts in the computer science community, two dis-tinct programming methods have evolved for parallel computing. One methodtakes advantage of the shared memory architecture by using compiler directivesfor parallelization, and the other uses explicit message passing libraries to moveinformation between distributed memories. In the former the OpenMP API hasevolved as the standard, whereas distributed memory programming has becomesynonymous with MPI (now Open MPI). The OpenMP API is a set of compiler



Processordomain

Globaldomain

GridBlock

Cacheblocks

Figure 6.14. Data structure and mapping to processors and cache memory.

directives that instruct the code at runtime to perform certain operations in parallelacross shared memory and can be implemented incrementally. Building parallelisminto a serial code with MPI on the other hand requires that the mode of parallelismbe decided a priori within a single-program multiple data (SPMD) framework suchthat for the most part the same operations are repeated on all processors. For CFDapplications, that usually translates to a decomposition of the spatial grid into con-tiguous pieces mapped to different processors with explicit message passing callsthrough MPI for transferring data from one processor to another for global syn-chronicity, and requires a major rewrite of the code. Whereas OpenMP is simpler toimplement and more flexible by being able to accommodate both different modesof parallelism and adaptation of parallelism [85, 86], it does not scale as well ascodes written in MPI for these same reasons. There is merit in combining the twoon DSM architectures to reap the benefits of both modes of parallelism (Tafti andWang [87]).

The multiblock framework provides a natural framework for parallelization.Figure 6.14 illustrates the data structure and the multiple levels of parallelism thatcan be extracted. Depending on the total number of mesh blocks and the degree ofparallelism sought, each processor can have multiple blocks residing on it as shown



Number of processors

Gflo

p/s

10.01

0.1

1

10

100

1000

10 100 1000 104

MPI-OpenMP (ASCI Red)MPI-OpenMP (ASCI Blue)

MPI (ASCI BLUE)

ASCI Blue Orgin-MIPS 250 MHz

ASCI Red Pentium Pro 200 MHz

Figure 6.15. Parallel scalability of ASPCG on two past ASCI platforms with MPIand MPI-OpenMP.

in Figure 6.14. The processor domain in Figure 6.14 is the same as an MPI process.Further within each block, “virtual cache blocks” are used for preconditioning thesolution of linear systems. The “virtual” blocks are not explicitly reflected in thedata structure, but are used only in the solution of linear systems. Hence embeddedin each MPI process are additional levels of parallelism across mesh blocks oracross cache blocks that can be exploited by OpenMP.

Figure 6.15 shows scalability studies on the ASCI Blue Origin 2000 clusterat Los Alamos and ASCI Red Intel machine at Sandia for a model problem. Thedomain size on each processor is maintained at 1,024 × 1,024 cells, with 1,024 sub-domains of size 32 × 32 for cache performance. The number of relaxation sweepsin the Jacobi smoother is kept at 40 in each block and 100 on the coarse level.The calculations on the ASCI Blue are run across a cluster of Ori-gins using only MPI (in and across Origins) or a combination ofMPI-OpenMP. In the hybrid MPI-OpenMP model, MPI processes are not onlyused across Origins but within Origins as well. For each nproc (total number ofprocessors used) in the figure, the multiple data points correspond to different com-binations of MPI and OpenMP threads executed across different number of Origins.On the ASCI Red, OpenMP is used within each two-processor node. In all cases,the scaled performance exhibits a perfect linear speedup showing the efficacy ofthe architecture as well as the use of hybrid parallelism.



6.11 Applications

The theory and modeling techniques described in this chapter have been used exten-sively for LES ([88–92]) and DES ([73–75, 93–95]) of turbine internal cooling ductswith ribs with rotation and centrifugal buoyancy effects at Re = 20,000 to 50,000and LES of leading edge film cooling at Re = 100,000 ([96–98]). A number ofinvestigations have also been performed in studying different fin configurations orheat transfer enhancement techniques in compact heat exchangers, which mostlyoperate in the transitional regime between Re = 100 to 2,000 ([99–103]).A dynamicmesh capability is presently being used to investigate flapping flight for applicationsto Micro-Air Vehicles (MAVs) at Reynolds numbers between 10,000 and 100,000([104, 105]). A two-phase dispersed phase capability using Lagrangian particledynamics has been used with LES to model fouling of internal cooling channelsand film cooling holes ([106, 107]).

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[90] Sewall, E. A., and Tafti, D. K. Large Eddy Simulation of Flow and Heat Transferin the 180o Bend Region of a Stationary Ribed Gas Turbine Blade Internal CoolingDuct, ASME J. Turbomach., 128, pp. 763–771, 2006.

[91] Sewall, E. A., and Tafti, D. K. Large Eddy Simulation of Flow and Heat Transfer inthe Developing Flow Region of a Rotating Gas Turbine Blade Internal Cooling Ductwith Coriolis and Buoyancy Forces, ASME J. Turbomach., 113(1), Jan. 2008.

[92] Elyyan, M. A., and Tafti, D. K. LES Investigation of Flow and Heat Transfer in aChannel with Dimples and Protrusions, ASME J. Turbomach., 130(4), 2008.

[93] Viswanathan, A. K., and Tafti, D. K. Detached Eddy Simulation of Turbulent Flowand HeatTransfer in a Ribbed Duct, ASME J. Fluids Eng., 127(5), pp. 888–896, 2005.

[94] Viswanathan, A. K., and Tafti, D. K. A Comparative Study of DES and URANS forFlow Prediction in a Two-Pass Internal Cooling Duct, ASME J. Fluids Eng., 128(6),pp. 1336–1345, Nov. 2006.

[95] Viswanathan, A. K., and Tafti, D. K. Detached Eddy Simulation of Flow and HeatTransfer in Fully-Developed Rotating Internal Cooling Channel with Normal Ribs,Int. J. Heat Fluid Flow, 27(3), pp. 351–370, 2006.

[96] Rozati, A., and Tafti, D. K. Large Eddy Simulation of Leading Edge Film CoolingPart-II: Heat Transfer and Effect of Blowing Ratio, ASME J. Turbomach., 130(4),2008.

[97] Rozati, A., and Tafti, D. K. Large-Eddy Simulation of Leading Edge Film Cooling,Int. J. Heat Fluid Flow, 29, pp. 1–17, 2008.

[98] Rozati, A., and Tafti, D. K. Effect of Coolant-Mainstream Blowing Ratio on LeadingEdge Film Cooling Flow and Heat Transfer – LES Investigation, Int. J. Heat FluidFlow, 29, pp. 857–873, 2008.

[99] Tafti D. K., and Cui, J. Fin-Tube Junction Effects on Flow and Heat Transfer in FlatTube Corrugated Multilouvered Heat Exchangers, Int. J. Heat Mass Transfer, 46(11),pp. 2027–2038, 2003.

[100] Elyyan, M. A., Rozati, A., and Tafti, D. K. Investigation of Dimpled Fins for HeatTransfer Enhancement in Compact Heat Exchangers, Int. J. Heat Mass Transfer, 51,pp. 2950–2966, 2008.

[101] Rozati, A., Tafti, D. K., and Blackwell, N. E. Effect of Pin Tip Clearance on Flow andHeat Transfer at Low Reynolds Numbers, ASME J. Heat Transfer, 130, July 2008.

[102] Srinivas, S., Tafti, D. K., Rozati, A., and Blackwell, N. E. Heat-Mass Transfer andFriction Characteristics of Profiled Pins at Low Reynolds Numbers, Numerical HeatTransfer A, 54(2), pp. 130–150, 2008.

[103] Elyyan, M., and Tafti, D. K. A Novel Split Dimple Interrupted Fin Configuration forHeat Transfer Augmentation, Int. J. Heat Mass Transfer, 52, pp. 1561–1572, 2009.

[104] Gopalakrishnan, P., andTafti, D. K.A Parallel Boundary Fitted Dynamic Mesh Solverfor Applications to Flapping Flight, Comp. Fluids, 38(8), pp. 1592–1607, 2009.

[105] Gopalakrishnan, P., and Tafti, D. K. Effect of Rotation and Angle of Attack on ForceProduction of Flapping Flights, AIAA J., 47(11), pp. 2505–2518, 2009.

[106] Shah, A., and Tafti, D. K. Transport of Particulates in an Internal Cooling RibbedDuct, ASME J. Turbomach., 129(4), pp. 816–825, October 2007.

[107] Rozati, A., Tafti, D. K., and Shreedharan, S. Effects of Syngas Ash Particle size onDeposition and Erosion of a Film-Cooled Leading Edge, Paper No. 56155, 2008ASME Summer Heat Transfer Conference, Aug. 10–14, Jacksonville, FL.


7 On large eddy simulation of turbulent flowand heat transfer in ribbed ducts

Bengt Sundén and Rongguang JiaLund University, Lund, Sweden

Abstract

This chapter presents some work on large eddy simulation (LES) for heat transferstudies. Firstly, the numerical accuracy is validated with the simulation of the fullydeveloped pipe flow with the direct numerical simulation (DNS) data. Secondly, thesolver is used to study the heat transfer and fluid flow in rib-roughened ducts, wherethe ribs could be straight or V-shaped. This is primarily to clarify a contradiction onwhich pointing direction is the better for V-shaped ribs. Lastly, current status andprospects of LES are presented.

Keywords: LES, Heat transfer, Ribs, Duct flow

7.1 Introduction

In order to allow the gas turbine designer to increase the turbine inlet temperaturewhile maintaining an acceptable material temperature, sophisticated cooling meth-ods are essential. For the internal cooling of the blades, ribbed ducts are usuallyused. The presence of ribs, also called roughness or turbulators, enhances the heattransfer coefficients by redevelopment of the boundary layer after flow reattach-ment between the ribs, and because of induced secondary flows. The rib layoutsand configurations can be varied and include 90-degree, 60-degree, and 45-degreeparallel ribs and 45-degree and 60-degree V-shaped ribs.

It is essential to be able to accurately predict the enhancement of heat transfergenerated by the roughness elements to ensure good design decisions. Accordingly,the heat transfer and fluid flow in ribbed ducts have been extensively studied bothexperimentally and numerically. A summary is presented in Jia et al. [1], wherecontradictions on experimental studies of V-shaped ribs have been identified. Theexperimental studies of both Han et al. [2] and Olsson and Sundén [3] show thatsquare ducts roughened with upstream pointing V-shaped ribs perform better thanthose pointing downstream. However, results on the contrary were obtained byTaslim et al. [4] and Gao and Sundén [5]. The contradiction might be due to the



differences in the experimental conditions, which need further clarification. In aprevious study [1] by the RANS method, side-wall heating was found to be themain reason for the discrepancy. However, the RANS method filtered out too manydetails, so that the explanation of the reason becomes difficult. In addition, theuncertainties of the RANS models may become large for complex geometries.Therefore, a more universal modeling method, large eddy simulation (LES), isemployed to further confirm and explain the findings from RANS.

LES is a computational method in which the large eddies are computed andthe smallest, subgrid-scale (SGS) eddies are modeled. The underlying premise isthat the largest eddies are directly affected by the boundary conditions, carryingmost of the anisotropic Reynolds stresses, and must be computed. The small-scaleturbulence is weaker, contributing less to the turbulent Reynolds stresses, is nearlyisotropic, and has nearly-universal characteristics. Therefore, it is more amenableto model.

In LES, the transport equations are filtered from the 3D, unsteady, Navier–Stokesequations by a filter function in space:

∂ρ

∂t+ ∂(ρ ui)

∂xi= 0 (1)

∂(ρ ui)

∂t+ ∂(ρ ujui)

∂xj= −

(∂p∗∂xi

− β)

+ ∂

∂xj

[µ

(∂ui

∂xj+ ∂uj

∂xi

)+ ρτij

](2)

∂(ρ φ∗)

∂t+ ∂(ρ ujφ∗)

∂xj= ∂

∂xj

(µ

Pr

∂φ∗∂xj

+ ρϕj

)− δ1jρ ujγ (3)

where τij = uiuj − uiuj are called the subgrid-scale (SGS) stresses, through whichthe small scales influence the large (resolved) scales. Accordingly, ϕj =φuj −φuj

are called the SGS heat fluxes. They are unknown and must be modeled by a SGSmodel.

The implicit model, proposed by Boris et al. [6], is the simplest SGS modelwithout introducing neither the filtering operation nor the SGS stress tensor. It isbased on the observation that truncation errors in such discretizations of Navier–Stokes equations introduce numerical dissipation with the implicit effects of thediscretization qualitatively similar to the effects of the explicit SGS models.

However, in the presence of the inviscid unsteadiness, the turbulence becomesmuch stronger, such as the separation zone downstream a rib. Consequently, thenumerical diffusion is not enough to dissipate the turbulent energy, especially fora fine grid. Some additional viscosity is necessary to prevent the buildup of turbu-lent energy at high frequency regions. Smagorinsky [7] was the first to postulatean explicit model for the SGS stresses. His model assumes the SGS stresses fol-low a gradient-diffusion process, similar to molecular motion. Consequently, τij isgiven by:

τij = 2νT Sij (4)


LES of turbulent flow and heat transfer 267

where Sij = 12

(∂ui∂xj

+ ∂uj

∂xi

)is called the “resolved strain rate,” and νT is the

Smagorinsky eddy viscosity given by:

νT = (CS)2√SijSij (5)

where and CS are the filter width (grid spacing) and the Smagorinsky constant,respectively, which must be specified prior to a simulation, varying from flow toflow. The Smagorinsky constant is set as CS = 0.1 in this simulation.

7.2 Numerical Method

The computations are carried out in an in-house parallel multi-block computercode CALC-MP [8, 9], based on the finite volume technique. The code uses acollocated body-fitted grid system and employs the improved Rhie and Chow inter-polation procedure method to calculate the velocities at the control volume faces.The SIMPLEC algorithm couples the pressure and velocity. An algorithm basedon SIP (Strongly Implicit Procedure) is used for solving the algebraic equations.Both the diffusion and convection terms are discretized by the second order centraldifference. The validation of the code has been presented by Jia [10].

7.3 Results And Discussion

7.3.1 Fully developed pipe flow

To illustrate the numerical accuracy of the present code, a fully developed pipe flowis calculated with LES, for Reτ = 460. Implicit model is used for modeling the SGSstresses, and a central scheme is used for the convection terms. A butterfly-typemulti-block grid is generated, and the approximate number of grid points is aroundone million.

Although flows such as a turbulent pipe flow are geometrically simple, theiraccurate simulation using techniques such as LES is in some way more difficultthan flows in complicated geometries. This is because no inviscid instability isavailable, its accurate simulation is an indication of the reliability of the method.

Figure 7.1a shows the spectrum at y+ = 230 (i.e., a point at the centerline of thepipe), and the cutoff of the high-frequency turbulence energy is clearly illustrated.

Figure 7.1b shows the mean main stream velocity and turbulent Reynoldsstresses by LES, in comparison with the DNS data of Eggels et al. [11]. Theyare in very good agreement. At the region close to the wall, the fluctuations in themain flow direction are much larger than those in the other two directions. Fromany RSTM model (Reynolds stress turbulence model), one can see that the produc-tion term for the fluctuations of the two cross directions is zero. The fluctuationsof these two directions obtain the turbulence energy from pressure strain terms.The pressure strain terms redistribute energy from the high-fluctuation directions



10−1

fDh/Ub

(a) Turbulent energy spectrum

100 101

−5/3

y+ = 230E = f −5/3

10−5

10−6

10−4

E(f

)

10−3

10−2

(b) Statistic Valuesr/D

U/u

τ

u ′v ′/u2u′/uτ

w ′/uτ

u ′/uτ

0.0

−1 −5

0.50

5

10

15

20

−0.50

1

2

3

4

LES DNS

(U′ iU

′ j)0.

5/u τ

0.5

−0.5−0.5

N/0

X0 0.5

(c) Instantaneous u′ from present LES

0

(d) Instantaneous u′ from DNS

π

π3

2π

1

2

33

0

0.5

3

0

0.5

3.5 4 4.5 5 5.5 6

3 3.5 4 4.5 5 5.5 6

(e) Contours of the instantaneous out-of-plane velocity at a center plane

Figure 7.1. The fully developed pipe flow case Reτ = 460.

to the low-fluctuation directions to reduce the difference. However, it can not takeaway the difference totally. Therefore, the mainstream direction fluctuation is muchlarger than those in the other two directions.

Figure 7.1c shows the main-flow direction instantaneous velocity fluctuations ina cross section of the pipe. One can clearly observe the ejection events from the walltoward the pipe center, and low-speed streaks will form downstream these ejection



(a) Geometry

P

Dh

Dh

e

(b) Block topology

plane-IV

plane-VI

plane-III

plane-II

plane-I

plane-V

plane-V YZ

X

plane-III

Figure 7.2. The computation geometry.

events. These ejection events and streaks produced the majority of the turbulence.Approximately, there are around six ejection events in the circumferential direc-tion. Normally the spanwise distance for the ejection events are around s+ = 100.Figure 7.1d shows a similar picture from the DNS of Eggels et al. [11].

Figure 7.1e shows the out-of-plane instantaneous velocity in a plane throughthe pipe centerline. Superimposed on top of the small-scale turbulence in the coreregion of the flow, large scale structure, on the scale of the pipe diameter, is clearlyseen. Near the pipe walls, the ejection events described earlier are seen as the darkand light regions that leave the pipe walls at an angle of approximately 20 degrees.

In general, we observed good agreement with the DNS data. This is true forboth mean and rms results.

7.3.2 Transverse ribbed duct flow

The computational geometry is schematically shown in Figure 7.2a, and the compu-tational block topology is illustrated in Figure 7.2b. The square duct with hydraulicdiameter Dh is roughened by periodically mounted ribs of size e, e/Dh = 0.1, and thepitch is 10e. The V-shape incline angle may be 90-degree, 60-degree, or 45-degree.When it is 90-degree, transverse ribs prevail. The simulated Reynolds number issmall, around 4,000, turbulent flow is ensured. The selection of a low Re is mainlybecause a relatively small cell number can be used, and the main features of the tur-bulence field are not different from the high-Re cases. In addition, the contradictionin the experiments is not Re-dependent.

Flow fieldFigure 7.3a shows the mean streamlines in the symmetry plane. The streamlinesare characterized by four recirculation zones, accompanying flow separation and



(a) Streamlinesx /e

y/e

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

(b) u ′/Ub

x /e

y/e

0.195

0.195

0.293

0.293

0.325

0.3250.358

0.3900.423

0.2930.260

0.260

0.228

0.195

0.455

0.2280.163

0.2600.260

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

0.228

0.228

0.4880.4550.4230.3900.3580.3250.2930.2600.2280.1950.1630.1300.0980.0650.033

(c) v ′/Ub

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

x /e

y/e

0.142

0.1600.178

0.213

0.249 0.2670.249

0.2310.213

0.1960.178

0.160

0.142 0.1780.1960.2130.2310.249

0.1240.160

0.107

0.142

0.178

0.160

0.142

0.124

0.1420.178

0.2130.249

0.1960.213

0.213

0.2670.2490.231

0.1960.213

0.1780.1600.1420.1240.1070.0890.0710.0530.0360.018

(d) w ′/Ub

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

x /e

y/e

0.124

0.1550.124

0.155

0.1860.217

0.248

0.279

0.248

0.2170.186

0.15

50.

124

0.311

0.37

3

0.279

0.2480.279

0.311

0.186

0.217

0.248

0.155

0.279

0.4660.4350.4040.3730.3420.3110.2790.2480.2170.1860.1550.1240.0930.0620.031

Figure 7.3. Streamlines and contours of the turbulent Reynolds stresses in thesymmetry plane (plane-I).



reattachment. The sudden expansion downstream of a rib leads to the generationof a wide recirculation vortex associated with a strong shear layer. Together withthis main recirculation, a second smaller separation is also identified, located inthe corner between the downstream face of the rib and the bottom wall. Afterreattachment, a new boundary layer grows; the flow is accelerated by the mainstream until it impinges upon the next rib. Finally, a fourth vortex is identified ontop of the rib.

Figure 7.3b shows contours of streamwise turbulence intensity u′/Ub. The peakvalues of u′/Ub are located at the rib top. This is due to acceleration and strongshear at the rib top. Some local maxima of u′/Ub values are found at the top of themain separation zone behind the upstream rib. They are related to the shear layeroriginating from the rib edge and located above the separation region. Contourswith high u′/Ub values are closer to the ribbed wall.

Figure 7.3c shows the y-direction velocity fluctuations. High v′/Ub values arefound above the main separation zone inside the cavity between two successive ribs.v′/Ub is small near the bottom wall in the cavity, indicating weak vertical turbulentmotions.

Figure 7.3d shows that the spanwise turbulence intensity, w′/Ub, is enhanced bythe ribs. The peak value lies at the lower corner in front of the rib, where stream wiseflow is blocked by the rib. A local maximum is also found in the main separationzone behind the upstream rib. The w′/Ub maximum is similar to the u′/Ub peak.High w′/Ub values are found upstream the rib, near the reattachment point in thecavity, and close to the rib front edge.

In general, very strong anisotropy is observed in the distribution of Reynoldsstresses in three directions. This makes it very difficult to predict it by using linear-eddy-viscosity models.

7.3.3 V-shaped ribbed duct flow

The solver was also employed to simulate the duct flow roughened with 60-degree-inclined V-shape ribs. The heat transfer coefficients are calculated locally, andcompared with the average value. The deviations are displayed with contours tohelp explain the discrepancy between different experimental results.

Figure 7.4 shows the deviation of the local Nu from the average Nu. The reasonfor the experimental contradiction can be explained from this picture. If we use onethermocouple (TC) to measure the temperature representing the surface; ideally,we should put theTC at the places, where the averaged value is located, i.e., the 0%-deviation contour lines. For the downstream pointing V-shaped ribs, if one puts theTC at the center of the surface, one will obtain a reasonably accurate average value(Figure 7.4a), because the value at the center of the surface is very close (around−5% underprediction) to the average value. For the upstream pointing V-shapedribs, however, if one puts the TC at the center of the surface, one will obtain anover predicted averaged value (Figure 7.4b), because the value at the center of thesurface is far (around 30% over prediction) from the averaged value.



50

50

−7

−14

07

4343

3636

36

29

29

29

21

21

21

1414

14

7

7

0

0−7−7

−21−29

−29

142129

3636

43

36

3629 21 14 7 0 −7

217

2 4x/e

z/e

x/e

z/e

−4

−2

0

2

4

−4

−2

0

2

4

6 8 10 12

2

(a) Downstream pointing V-shaped ribs

(b) Upstream pointing V-shaped ribs

4 6 8 10 12

Figure 7.4. Deviation (%) from the average Nu.

With the liquid crystal (LC) method, the numbers of the sampling points forheat transfer coefficients are doing much better to correctly represent anisotropicheat transfer behavior. Therefore, correct conclusions are more likely obtained.

In summary, we can conclude that the LC experimental data in [4] and [5] aremore reliable than the thermocouple data in [2] and [3]. The main reason is that theprecise temperature distribution is important. The downstream pointing V-shapedribs are providing better heat transfer efficiency.



7.4 Conclusions

The LES method implemented in the present solver has good accuracy forpredictions of heat transfer and fluid flow.

Based on the simulation for the V-shaped ribbed duct flow, we have found thatthe LC experimental data are more reliable than those of TC data, which is thereason for the discrepancy of the experimental conclusions. It further confirmedthat the V-shaped ribs pointing downstream are performing better in heat transferenhancement.

However, for general application of LES in heat transfer simulations, there areremaining difficulties:

1. Treatment of near-wall turbulence: The near-wall turbulence is highly turbu-lent and small scaled (both in time and space), and plays a too significantrole in heat transfer predictions to be neglected. These small scales normallycannot be captured by the simulations oriented for large scales. Therefore,subgrid models are vital here. Consequently, SGS models would be very impor-tant for heat transfer problems. Some numerical error may play a role here,but may only have limited success. Some SGS models seem promising, suchas WALE (wall-adapting local eddy-viscosity) and the dynamic Smagorinskymodel (DSM).

2. Computational cost: It takes long time to obtain enough LES statistic samples,which imposes a heavy burden on the computational speed and capacity. SomeRANS calculations may be finished in minutes, but it may take days for acomparable LES calculation.

With the development of even faster and larger computer and quicker commu-nication hardware, at the same time dipping prices, these computational costs willbecome less significant, but the software level parallelization is still somewhat leftbehind.

In general, we can foresee advancement of computational power and conse-quently the SGS model will become less important at the end, since mesh can befiner and time steps can be smaller, and gradually come close to the DNS level,although the better understanding of near wall turbulence will speedup this proce-dure. Robust and accurate schemes and algorithms will always be an important partof research.

As a summary, on today’s parallel computer clusters built with commodity com-ponents, realistic heat transfer problems can be resolved with LES at an acceptablecost and decent accuracy, while the advancement of the computational technologieswill make the solution faster and more accurate.

Acknowledgment

The Swedish Energy Agency financially supported this research work.



References

[1] Jia, R., Saidi, A., and Sundén, B. Heat transfer enhancement in square ducts withV-shaped ribs, ASME J. Turbomachinery, 125(4), pp. 788–791, 2003.

[2] Han, J. C., Zhang, Y. M., and Lee, C. P. Augmented heat transfer in squarechannels with parallel, crossed, and V-shaped angled ribs, ASME J. Heat Transfer,113, pp. 590–596, 1991.

[3] Olsson, C. O., and Sundén, B. Experimental study of flow and heat transfer inrib-roughened rectangular channels, Exp. Therm. Fluid Sci., 16(4), pp. 349–365, 1998.

[4] Taslim, M. E., Liu, T., and Kercher, D. M. Experimental heat transfer and friction inchannels roughened with angled, V-shaped, and discrete ribs on two opposite walls,ASME J. Turbomachinery, 118(1), pp. 20–28, 1996.

[5] Gao, X., and Sundén, B. Heat transfer and pressure drop measurements in rib-roughened rectangular ducts, Exp. Therm. Fluid Sci., 24(1–2), pp. 25–34, 2001.

[6] Boris, J. P., Grinstein, F. F., Oran, E. S., and Kolbe, R. L. New insights into large eddysimulation, Fluid Dyn. Res., 10, pp. 199–208, 1992.

[7] Smagorinsky, J. General circulation experiments with the primitive equations. I. Thebasic experiment, Mon. Weather Rev., 91, pp. 99–164, 1963.

[8] Jia, R., and Sundén, B.A Multi-block implementation strategy for a 3D pressure-basedflow and heat transfer solver, Numer. Heat Transfer B, 44(5), pp. 457–472, 2003.

[9] Jia, R., and Sundén, B. Parallelization of a multi-blocked CFD code via threestrategies for fluid flow and heat transfer analysis, Comput. Fluids, 33, pp. 57–80, 2004.

[10] Jia, R., Turbulence modelling and parallel solver development relevant for investiga-tion of gas turbine cooling processes, Doctoral thesis, Department of Energy Sciences,Lund Institute of Technology, 2004.

[11] Eggels, J. G. M., Unger, F., Weiss, M. H., Westerweel, J., Adrian, R. J., Friedrich, R.,and Nieuwstadt, F. T. M. Fully developed turbulent pipe flow: A comparison betweendirect numerical simulation and experiment, J. Fluid Mech., 268, pp. 175–209, 1994.


8 Recent developments in DNS and modelingof turbulent flows and heat transfer

Yasutaka Nagano and Hirofumi HattoriDepartment of Mechanical Engineering, Nagoya Institute ofTechnology, Japan

Abstract

The purpose of this chapter is to review recent studies on direct numerical simu-lation (DNS) and turbulence models from the standpoint of Computational FluidDynamics (CFD) and Computational Heat Transfer (CHT). Trends in recent DNSresearch and its role in turbulence modeling are discussed in detail. First, the instan-taneous and the Reynolds-averaged governing equations for flow and energy arewritten in rotation and nonrotation frames. Secondly, as a typical recent DNS usingthe high-accuracy finite-difference method, the detailed DNS results of velocityand thermal fields in turbulent channel flow with transverse-rib roughness are pre-sented. The problem is very important for elucidating the structure, which affectsheat transfer enhancement. Thirdly, our recent DNS of rotating channel flows withvarious rotating axes and heat transfer is described. Rotating flows with heat trans-fer are encountered in many applications relevant to engineering, such as in gasturbines. This DNS was conducted using the spectral method. Therefore, two sig-nificant types of DNSs are presented in this chapter. Finally, on the basis of DNSs,much emphasis is placed on evaluating the performance of various kinds of turbu-lence models used in the velocity and thermal fields. Then, a new turbulence modelis proposed for the accurate prediction of rotating turbulent flow with heat transfer.We should, however, bear in mind that a universal turbulence model can never beestablished without further development of DNS.

Keywords: Turbulence, Heat Transfer, DNS, Turbulence Model, Wall Roughness,Rotation

8.1 Introduction

Various attempts to analyze turbulence phenomena using available computer tech-niques have been made by many researchers. During the past three decades, directnumerical simulation (DNS) of three-dimensional time-dependent turbulent flows



has become a powerful tool for in-depth investigation of the structure of turbu-lent shear flows. This progress is mainly due to the considerable improvement incomputational power of supercomputers and the great efficiency of numerical meth-ods. With the advent of new massively parallel computers, this computational poweris still continuously increasing and DNS of moderately high Reynolds number flowsin complex geometries has been becoming a reality. One distinct advantage of theDNS technique is ready and accurate information such as the pressure fluctuation pand dissipation rate εof turbulent kinetic energy k , which can never be obtained fromthe measurements that have played the central role in traditional turbulence research.

DNS has also contributed to the development of turbulence models. Especially,two-equation (i.e., k − ε for a velocity field and kθ − εθ for a thermal field, wherek and ε are the turbulent kinetic energy and its dissipation rate, and kθ and εθthe intensity of fluctuating temperature and its dissipation rate, respectively) andsecond-moment closure (i.e., Reynolds stress and turbulent heat flux) models havemade rapid progress with the aid of DNS. It is well known that the standard two-equation and second-moment closure models provide comparatively satisfactoryresults for simple flow fields, whereas in more complicated flow results are not assatisfactory as initially expected. Thus, constructing a more sophisticated turbu-lence model applicable to various types of flows is the central issue.

In this chapter, we first discuss the trend in recent DNS research and its applica-bility to turbulence model construction or evaluation. Then, we show two types ofDNSs, one is conducted using the highly accurate finite-difference method and theother performed using the spectral method. Both DNSs mimic complex turbulentflows with a high degree of accuracy. And finally, we intend to introduce our latestturbulence models used in the velocity and thermal fields of technological interest.In addition, we introduce some new methods of evaluation and construction ofturbulence models and attempt to comment on future research.

8.2 Present State of Direct Numerical Simulations

At present, DNS is widely regarded as a new method in place of experiments usingapparatus. Since various information that can never be obtained from experimentsis immediately and accurately supplied by DNS, DNS is expected to help establishnew turbulence theories, analyze turbulence phenomena, and construct universalturbulence models. To establish turbulence theories and analyze turbulence phe-nomena, Robinson [1] conducted thorough investigations using the DNS databaseon boundary layer flow. Previous DNS studies were compiled and reviewed byKasagi [2], especially the relation between velocity and thermal fields.

From the viewpoint of turbulence modelers, there is much more interest in therole of DNS in constructing a universal turbulence model. For example, detailedinformation on fundamental turbulence quantities such as the pressure fluctuationp and the dissipation rate ε have brought to light many problems inherent to theexisting turbulence models and paved the way for the construction of a universalturbulence model. Also, information on budget profiles of various transport equa-tions have encouraged active modeling of the elemental processes of turbulence [3].


Recent developments in DNS 277

As a result, DNS has emerged as the most important tool for the development ofturbulence model.

On the other hand, some new attempts to construct the turbulence model withthe aid of the DNS databases have been suggested. Nagano and Shimada [4],for example, have devised a method for evaluating a modeled-transport equationitself using DNS data. This method also has the advantage of individually estimatingelemental processes of turbulence.

Early DNS research mainly aimed to consolidate the foundation of numericaltechniques under simple flow conditions, e.g., homogeneous shear flow [5], two-dimensional channel flow [6, 7], and turbulent boundary layer flow [8]. RecentDNS research focuses on the analyses of turbulence phenomena (e.g., mechanismsof turbulence generation and destruction, the interaction between inner and outerlayers, and the interaction between velocity and thermal fields). Moreover, DNShas made it possible to construct databases under more complicated flow condi-tions. Examples include adverse-pressure-gradient boundary layer flow [9, 10],backward-facing step flow [11], channel flow with riblets [12, 13], rotating channelflow [14]–[18], square duct flow [19], channel flow with injection and suction [20],channel flow under stable and unstable stratifications [21], impinging jet [22], andstable and unstable turbulent thermal boundary layer [23], all of which introducehighly advanced DNS techniques. DNS databases presented above are now availablefor assessing various kinds of turbulence models.

8.3 Instantaneous and Reynolds-Averaged GoverningEquations for Flow and Heat Transfer

In DNS of turbulent motion, the three-dimensional unsteady Navier–Stokes equa-tions for an incompressible fluid are solved directly for the instantaneous values ofthe velocity components and pressure. Here, we consider a turbulent heated chan-nel flow with rotation at a constant angular velocity as shown in Figure 8.1. Thegoverning equations for an incompressible rotating channel flow in reference framerotational coordinates can be described in the following dimensionless forms:

∂u+i

∂x+i

= 0 (1)

Du+i

Dt+= −∂p

+eff

∂x+i

+ 1

Reτ

∂2u+i

∂x+j ∂x

+j

− εikRoτku+ +∗

1 (2)

Dθ+

Dt+= 1

Pr Reτ

∂2θ+

∂x+j ∂x

+j

(3)

where t+(= tuτ/δ) is the time, u+i (= ui/uτ) is the instantaneous velocity in

x+i (= xi/δ) direction, θ+(= θ/ ) is the instantaneous temperature. These are

normalized by the friction velocity, uτ , the channel half width, δ, and the temper-ature difference, (= H − c), between upper ( c) and lower ( H ) walls.



Cooled wall

Heated wall

x3

x2

x1

2δΘc

ΘH

Ω3

Ω2

Ω1

Flow

Figure 8.1. Heated rotating channel with arbitrary rotating axes.

The Reynolds-, the rotation-, and Prandtl numbers are respectively defined asReτ = uτδ/ν, Roτk = 2kδ/uτ , and Pr = ν/α, where ν is the kinematic diffusivityfor momentum, k is the angular velocity with xk axis and α is the thermal dif-fusivity for heat. The centrifugal force can be included in the effective pressurep+

eff (= p+s − Ro2r2

c/8), if fluid properties are constant, where p+s is the normal-

ized static pressure, Ro = (RojRoj)1/2 is the absolute rotation number, and rc isthe dimensionless distance from the rotating axis [17]. ∗

1 is the nondimensionalconstant mean pressure gradient to maintain a flow, which is implemented only fora flow with asymmetrically roughened walls at Ro = 0 to make calculations morestable. As described later, the spectral method [16] is used for the DNSs of rotatingchannel flows. The computational conditions are listed in Table 8.1. The bound-ary conditions are non-slip conditions for the velocity field and different constanttemperatures for the thermal field on the walls, = constant, as well as periodicconditions in the streamwise and spanwise directions.

In turbulence modeling, the so-called Reynolds-averaged equations are usu-ally employed, and turbulence models based on Reynolds-averaged Navier–Stokesequation are often referred to as RANS models. The governing equations for RANSturbulence model can be written as follows [24]:

∂U i

∂xi= 0 (4)

DU i

Dt= − 1

ρ

∂Peff

∂xi+ ∂

∂xj

(ν∂U i

∂xj− uiuj

)− 2εikkU (5)

D

Dt= ∂

∂xj

(α∂

∂xj− ujθ

)(6)

uiuj = 2

3kδij − 2C0νtSij + Nonlinear terms (7)

ujθ = −αtjk∂

∂xk+ Nonlinear terms (8)

where C0 is a model constant, k(= uiui/2) is turbulent kinetic energy,Sij[= (∂U i/∂xj + ∂U j/∂xi)/2] is the strain tensor, U i is the mean velocity in xi



Table 8.1. Methods for direct numerical simulation

Grid Regular grid

Time advancementViscosity term Crank–Nicolson methodOther terms Adams–Bashforth method

Spatial scheme Spectral methodGrid points (x1 × x2 × x3) 64 × 65 × 64Computational volume 4πδ× 2δ× 2πδ

direction, αtjk = −Ct0ujukτm is the anisotropy eddy diffusivity for heat, is the

mean temperature, νt = Cµ fµ(k2/ε) is the eddy diffusivity for momentum, ρ isdensity, and τm is the hybrid/mixed time scale.

The transport equations of turbulence quantities, i.e., turbulent kinetic energy, k ,dissipation rate of turbulent kinetic energy, ε, temperature variance, kθ(= θ/2), anddissipation rate of temperature variance, εθ , which compose the eddy diffusivitiesare given as follows [16]:

Dk

Dt= ν ∂

2k

∂xj∂xj+ Tk +k + Pk − ε (9)

Dε

Dt= ν ∂

2ε

∂xj∂xj+ Tε +ε + ε

k(Cε1Pk − Cε2 fεε) + E + R (10)

DkθDt

= α ∂2kθ

∂xj∂xj+ Tkθ + Pkθ − εθ (11)

DεθDt

= α ∂2εθ

∂xj∂xj+ Tεθ + εθ

kθ(CP1 fP1Pkθ − CD1 fD1εθ)

(12)+εθ

k(CP2 fP2Pk − CD2 fD2ε) + Eθ + Rθ

where Tk , Tε, Tkθ , and Tεθ are turbulent diffusion terms, k and ε are pressurediffusion terms, and Pk (= −uiuj∂U i/∂xj) and Pkθ (= −ujθ∂ /∂xj) are productionterms, E and Eθ are extra terms to correct the near-wall behaviors of ε and εθ, and Rand Rθ are rotation-influenced additional terms. Note that, in the rotational coordi-nate system, the vorticity tensor,ij[= (∂U i/∂xj −∂U j/∂xi)/2], should be replacedwith the absolute vorticity tensor in equations (7) and (8), i.e., Wij =ij + εmjim

to satisfy the material frame indifference (MFI) [25].

8.4 Numerical Procedures of DNS

In general, the DNSs are performed using the two methods described brieflyin the following. Each method has its own special feature. Thus, between thetwo approaches one should decide which to choose after careful consideration ofproblems to be solved.



Cooled wall Tw = Tc

Heated wall Tw = TS

y

0

zS

W

H

x

4.8δ

3.2δ

2δ

Figure 8.2. Channel with transverse-rib roughness and coordinate system.

8.4.1 DNS using high-accuracy finite-difference method

To solve a flow problem in complex geometries, a high-accuracy finite-differencemethod is the best choice. Figure 8.2 shows a schematic of the channel withtransverse-rib roughness and the coordinate system used in the present study [13].This type of flow is definitely complicated. The origin of the coordinate axes islocated in the middle of the enclosure between the ribs. Table 8.2 summarizessome details of the high-accuracy finite-difference method. DNS was carried outwith a constant mean pressure gradient to balance the wall shear stress on bothwalls. Note that a fully consistent and conservative finite-difference method is usedfor the convective term of the Navier–Stokes equation [26]. In order to solve thePoisson equation for the pressure, the standard SOR method is applied and theover-relaxation coefficient is set to 1.5. The adequate numbers of grid points arearranged in each direction for DNS so as to exactly capture the turbulent heattransfer as listed in Table 8.2. Especially, the grid resolution of spanwise direction,z∗, must be arranged smaller than 10.0. The time advancement ist∗ = 1 × 10−4,and the total time steps needed for the statistical quantities to converge reasonablyare 300,000. The numerical scheme used in this study is validated by comparingthe statistical quantities, including the budget of the turbulent kinetic energy, inplane channel flow with those calculated by Nagano and Hattori [16], employing aspectral method as described next.

8.4.2 DNS using spectral method

Referring to Kim et al. [6], a fourth-order partial differential equation for v (wall-normal velocity component), a second-order partial differential equation for thewall-normal component of vorticity, and the continuity equation are solved toobtain the instantaneous flow field. The numerical method is fully described byKim et al. [6]. That is, a spectral method is adopted with Fourier series in thestreamwise and spanwise directions and Chebyshev polynominal expansion in thewall-normal direction. The collocation grid used to compute the nonlinear termsin physical space has 1.5 times finer resolution in each direction to remove alias-ing errors (sometimes referred to as padding or 3/2-rule). For time integration,



Table 8.2. Computational methods

Channel with transverse-rib roughness

Grid Staggered gridCoupling algorithm Fractional step methodTime advancement

Conductive term Crank–Nicolson methodOther terms Adams–Bashforth method

Spatial scheme 2nd-order central differenceComputational volume 4.8δ× 2δ× 3.2δGrid points 192 × (96 + 36 or 18) × 96(x × y (rib outside + rib inside) × z)Grid resolution x∗ = 3.75

y∗ = 0.30 ∼ 6.50z∗ = 5.0

the second-order Adams–Bashforth and Crank–Nicolson schemes are used for thenonlinear and viscous terms, respectively (see Section 4.1).

8.5 DNS of Turbulent heat Transfer in Channel Flow withTransverse-Rib Roughness: Finite-Difference Method

DNS of turbulent heat transfer in channel flow with transverse-rib roughness wascarried out using the high-accuracy finite-difference method [13], in which flowparameter and five types of wall roughness are arranged as listed in Table 8.3. Thenumber of grid points in the y-direction in the enclosure is 36 for Cases 1, b, and c,and 18 for Cases 2 and 3 as indicated in Table 8.2. According to the classification ofroughness [27, 28], Case 1 in Table 8.3 belongs to k-type roughness, and is similarshape to the experiment of Hanjalic and Launder [29]. In Case 2, the ribs are set tohalf the height of Case 1, and in Case 3, the height is set to half the height of Case 2.In the k-type roughness, the effect of the roughness is expressed in terms of theroughness Reynolds number, H+ = uτH/ν. In Cases b and c, the height of the ribsis set the same as in Case 1, though the spacing of the ribs is varied systematically.Case c belongs to d-type roughness, in which the effect of the roughness cannot beexpressed by H+. On the other hand, Case b is not regarded as the exact d-type, butd-like type roughness. Note that the two kinds of roughness are extreme versionsand intermediate forms can exist [28].

8.5.1 Heat transfer and skin friction coefficients

Figures 8.3 and 8.4, respectively, show the skin friction coefficient and the Nusseltnumber against the Reynolds number. By taking into account the asymmetry [30]



Table 8.3. Flow and rib parameters

Reτ 150

Pr 0.71

H W S Roughness type

Case 1 0.2δ 0.2δ 0.6δCase 2 0.1δ 0.2δ 0.6δ k typeCase 3 0.05δ 0.2δ 0.6δCase b 0.2δ 0.4δ 0.4δ d-like typeCase c 0.2δ 0.2δ 0.2δ

103

0.01

0.05

Cfr,

Cfs

Cf = 12Rem−1 Cf = 0.073Rem

−1/4 : Dean (1978)

104

Rem

Case 1Smooth

Case 2Case 3Case bCase c

Plane channel

Rough

Figure 8.3. Distributions of skin friction coefficients.

2 × 103

10

104

50

Redh

NU = 0.021Pr 0.5Redh0.8

Case 1Rough Smooth

Case 2Case 3Case bCase c

Plane channel

: Kays & Crawford (1993)

Nu r

,Nu s

Figure 8.4. Distributions of Nusselt numbers.



in the flow field disturbed by the ribs, the skin friction coefficient and the Nusseltnumber at the rough wall are defined as follows:

Cf r = 2 τwr

ρ〈U 〉2r, Nur = 2 qwr d

(Th − 〈T 〉r) λ(13)

where d indicates the distance between the wall and the location where the meanvelocity becomes maximum, and 〈〉 denotes the bulk mean over the distance d. Atthe smooth wall, they are similarly defined as follows:

Cfs = 2τws

ρ〈U 〉2s, Nus = 2qws(2δ− d)

(〈T 〉s − Tc) λ(14)

The wall shear stress, τws, at the (upper) smooth wall is estimated directly bythe mean velocity gradient on the wall and then, τwr at the (lower) rough wall iscalculated from the balance of the imposed pressure gradient:

τwr = −2δdp

dx− τws = 2τw0 − τws (15)

The above-defined wall shear stress at the rough wall, τwr, includes the pressuredrag (form drag) as well as the viscous drag:

τwr = τwp + τwv (16)

On the other hand, because the fully developed state is assumed for the thermalcondition in the present study, there is no increase in the enthalpy of the flowingfluid in the streamwise direction. Thus, the overall heat flux at the rough wall, qwr, isequal to the absolute value of that at the smooth wall, qws, which can be estimateddirectly by the mean temperature gradient on the wall. In order to compare theresults with the correlation curve by Kays and Crawford [31] for the pipe flow, thelength scales of the Reynolds and Nusselt numbers are doubled in Figure 8.3. Inthose figures, the DNS results in the plane channel flow (Reτ = 150) calculated bythe spectral method [16] are also included to compare with the results of the presentDNS using the high-accuracy finite-difference method.

From Figures 8.3 and 8.4, it can be seen that the skin friction coefficient and theNusselt number at the rough wall become very large in comparison with those atthe corresponding smooth wall, where these quantities are well represented by thecorrelation curves. Moreover, from Figure 8.3, even though the imposed pressuregradient is the same as in the smooth wall, the rib decreases the bulk mean velocity,so the Reynolds number heavily depends on the rib configuration; in Case 1, wherethe rib is highest in the k-type roughness walls, the Reynolds number is the smallestamong them. In the present study, Case 3 is found to be the lowest in drag. Onthe other hand, in the d-like type roughness (Cases b and c), the heat transferaugmentation is smaller than that in the k-type roughness with the same roughnessheight (Case 1).



Table 8.4. Coefficients for heat transfer rate at rough wall

(Str/Cf r)/(St0/Cf 0) Cf r/Cf 0 Nur/Nu0

Case 1 0.61 3.93 2.41Case 2 0.71 2.87 2.04Case 3 0.85 1.66 1.42Case b 0.69 2.44 1.67Case c 0.69 2.10 1.46

Table 8.4 shows the Stanton number, the skin friction coefficient, and the Nusseltnumber, which are divided by the estimated smooth-wall values from the respectivecorrelation curve [31, 32] for the same Reynolds number. In the k-type classifica-tion, Case 1 enhances heat transfer most. However, from the viewpoint of the heattransfer characteristic including the drag, if we compare the Stanton number dividedby the skin friction coefficient Str/Cf r , from Table 8.4, Case 1 enhances the heattransfer more than the smooth wall, but the overall characteristics of heat transfer donot improve because of the large drag. However, Case 3, the k-type with the lowestribs, promotes the heat transfer with very low drag. This case is the most efficientfrom the standpoint of overall heat transfer performance including the drag. In thed-like type roughness, the heat transfer characteristic cannot be improved regardlessof the rib spacing. The stagnation region in the enclosure becomes larger than thatin the k-type roughness (not shown here), and the deterioration in heat transfer hasbeen profound there. Thus, in the following, we discuss the effects of the height ofthe rib on the statistical characteristics of thermal property in the k-type roughness,which includes a rib configuration which promotes the heat transfer.

8.5.2 Velocity and thermal fields around the rib

To visualize the velocity and thermal fields around the rib, Figures 8.5 and 8.6 showthe mean streamlines estimated from the mean velocity profiles and contour linesof the mean temperature distributions. The averages are taken only with respectto the spanwise direction in these figures. Two-dimensional vortices exist in theenclosures between the ribs in Cases 1 and 2, and their shapes are different in eachcase. In Case 1, the center of the two-dimensional vortex is biased to the upstreamside of the rib and becomes asymmetric, whereas in Case 2, the center of the vortexis biased to the downstream side of the rib. Moreover, in Case 3, the small two-dimensional vortices are located on both sides of the rib, and the flow reattachmentis seen in the enclosure.

On the other hand, from the contour lines of the mean temperature, around thefront corner of the rib, the spacing between the lines becomes smaller, making forvery active heat transfer there. In the enclosure between the ribs, the contour linesare distorted corresponding to the streamlines and are not parallel to the bottomwall. Especially, in Case 3, it is confirmed that the mean temperature contour lines



Case 1

Case 2

Case 3

Figure 8.5. Streamlines of mean velocity averaged in the spanwise direction. Flowis left to right.

Case 1

Case 2

Case 3

Figure 8.6. Contour lines of mean temperature averaged in the spanwise direc-tion. The interval between successive contour level is 0.02. Flow is leftto right.

are densely distributed, causing more enhanced heat transfer. On the upstream sideof the ribs, the wavy patterns of temperature can be seen. This phenomenon is acharacteristic specific to a temperature field near the upstream side of the rib.

Figures 8.7 and 8.8 show the local skin friction coefficient, Cfr(x)[= 2τwv(x)/ρ〈U 〉2

r ] and the local Nusselt number, Nur(x)[= 2qwr(x)d/(Th − 〈T 〉r)λ] along thebottom of the enclosure and the crest on the rough wall, divided by the values on thesmooth wall. These mean quantities are obtained by averaging with respect to time,the spanwise direction and the streamwise period of the rib roughness. The localwall shear stress, τwv(x), and wall heat flux, qwr(x), are estimated directly fromthe gradients of mean velocity and temperature, respectively. Thus, the local shearstress shown in Figure 8.7 does not include the pressure drag see equation (16).By averaging the local wall shear stress along the bottom of the enclosure and thecrest in the x-direction, the contributions of the viscous drag to the total drag ofthe rough-wall side are estimated at 4.5%, 1.1% and, 20.8%, in Cases 1, 2, and 3,respectively. Ashrafian and Andersson [33] reported the contribution of the viscousdrag to the total drag was 2.5%, in their DNS calculation of the rib-roughenedchannel flow (the pitch to the height ratio, (S + W )/H , is 8, Reτ0 = 400). Leonardiet al. [34] have systematically investigated the effects of the pitch to the heightratio on the contributions from the viscous and pressure drags. They reported that



Case 1 H = 0.2δCase 2 H = 0.1δCase 3 H = 0.05δ

Enclosure Rib

Cfr

/Cf0

−0.2−4

0

4

0 0.2 0.4x/d

Figure 8.7. Profiles of local skin friction coefficients around rib along the horizontalwall with x = 0 at the middle of the enclosure.


Enclosure Rib

Nu

r/N

u0

−0.2

0

5

0 0.2 0.4x/d

Figure 8.8. Profiles of local Nusselt number around rib along the horizontal wallwith x = 0 at the middle of the enclosure.

the total drag is determined almost entirely by the pressure drag when the ratio iswithin the range from 6 to 20.

From Figure 8.7, the local skin friction coefficient becomes minimum in theenclosure and maximum at the front corner of the rib in Cases 1 and 2. On theother hand, in Case 3, although the local skin friction coefficient takes the maxi-mum at the front corner of the rib, both a large overall decrease and a local peakin the enclosure cannot be seen; but it changes sign in the enclosure. The reattach-ment point is located at x/δ= 0.04; thus the forward flow region is observed over0.04< x/δ< 0.21 until separation occurs in the enclosure between the ribs.

The local Nusselt numbers in Cases 1 and 2 distribute similarly to the abso-lute value of the local skin friction coefficient. However, in Case 3, the Nusseltnumber increases over the entire region in the enclosure. This situation is alsoobserved near the reattachment region of the backward-facing step flows [35], andit is reported that the heat transfer coefficient reaches maximum there. Similarly,



0.5

0.3

0.1

0

0.1 0.20−0.1

U/U

m

0.5

1.5

1

0.1 0.20 00

1 2

Rough wall Smooth wall

Plane channel

Case 2 H = 0.1δCase 1 H = 0.2δ

Case 3 H = 0.05δ

y/δ

Ret0 = 150x/δ = 0

Figure 8.9. Profiles of mean velocity.

the local Nusselt number increases in approaching the reattachment point x/δ= 0.04and the following forward flow region (0.04< x/δ< 0.21). From the above results,it is confirmed that in Case 3, the heat transfer is promoted, yet with only a relativelysmall increase in the drag.

8.5.3 Statistical characteristics of velocity field and turbulent structures

In a velocity field of a channel flow with a rib surface, the flow motions in theenclosure between the ribs strongly affect the flow field above the ribs. We examinedthe various ways of averaging, e.g., average over the x–z plane, spanwise average atthe middle of the enclosure (x/δ= 0) and at the rib crest (x/δ= 0.4). The differenceswere not seen along the streamwise direction except near the roughness element.Thus, in the following, the statistical quantities of turbulence in the middle of theenclosure (x/δ= 0) are discussed.

Figure 8.9 shows the mean velocity normalized by the bulk velocity Um. Becauseof the effects of the rib, the velocity decreases on the rough-wall side. The corre-sponding distributions of the Reynolds shear stress and turbulent kinetic energy areshown in Figures 8.10 and 8.11. As the rib height increases, turbulence is promoted,and both the Reynolds shear stress and the turbulent kinetic energy increase nearthe wall. This affects the region over the center of the channel. However, in thenear-wall region of the opposite wall, there is only a small effect, in comparisonwith the results for the plane channel flow. Despite the negative value of the velocitygradient near the wall in the enclosure as seen in Figure 8.9, the Reynolds shearstress takes a positive value, thus confirming the occurrence of counter gradientdiffusion. Because the production of the Reynolds shear stress is mainly maintainedwith the pressure–strain correlation and the pressure diffusion, no production fromthe mean shear is observed. An experimental study [29] indicated that the placewhere the Reynolds shear stress becomes zero and the mean velocity becomesmaximum is different. However, the difference between them is very small; 2.0%,2.8%, and 0.8% of a channel width for Cases 1, 2, and 3, respectively.



−0.005

0

0 1

y/δ

2

0.005

u

n/U

m20.01

0.015Case 1 H = 0.2δCase 2 H = 0.1δCase 3 H = 0.05δPlane channel

Ret0 = 150

x /δ = 0

Figure 8.10. Profiles of Reynolds shear stress.

0.01

0.02

0 1

y/δ

k/U

m2

2

0.03

0.04

0.05Case 1 H = 0.2δCase 2 H = 0.1δCase 3 H = 0.05δPlane channel

Ret0 = 150

x /δ = 0

Figure 8.11. Profiles of turbulent kinetic energy.

Figure 8.12 shows the turbulence intensities normalized by the friction velocityon the rough wall. In all cases, the statistical quantities get together well in theupper region of the ribs. However, from Figure 8.11, the turbulence intensitiesincrease relative to the bulk mean velocity. The budget of the turbulent kineticenergy is shown in Figure 8.13. In Case 1, the convection term is enhanced incomparison with the plane channel flow. This is because the mean vertical velocitycomponent near the rib is increased, and also there is a variation in the turbulencein the streamwise direction. Moreover, the contributions from the turbulent andpressure diffusion terms are very large; in the enclosure between the ribs, there isno turbulence production from the mean shear, but turbulent transport maintainsthe turbulence there. In Case 3, the distribution of the budget is similar to the resultin the plane channel, in comparison with Case 1, but the contributions from theturbulent transport (turbulent and pressure diffusions) are large.



3Rough wall Smooth wall

Case 1Case 2Case 3Plane channel

Ret0 = 150

v+ rm

sw

+ rms

u+ rm

s2

1

1

1

0.5

1.5

0.5

100 200

y+

00

0

0

Figure 8.12. Distributions of turbulence intensities.

Next, to investigate the anisotropy of turbulence, Figure 8.14 shows thepressure–strain correlation terms see equation (23) which are important in the near-wall region. The pressure–strain correlation is affected because the flow impingeson the upstream side of the rib wall, and the pressure fluctuations increase over awide region on the rough-wall side in comparison with the smooth-wall side. If wecompare Case 1, where the rib is highest, with the plane channel flow, the respectivecomponents of the pressure–strain term become maximum in the region betweenthe ribs, and a very active energy exchange occurs. In Case 1, the sign of 11becomes positive at y+ 15. In ordinary wall turbulence without external force,it is observed that in the near-wall region, the splatting phenomenon caused by v2

gives the energy to w2 as seen in the DNS results [16]. However, in the enclosurebetween ribs, the energy does not redistribute to the w2 component. This can beattributed to the two-dimensional vortices existing in the enclosure, which promote



Plane channelProductionDissipation

Turbulent diffusionViscous diffusion

Convection

In enclosure(cavity)

Case 1 H = 0.2δ

Case 3 H = 0.05δ

Pressure diffusionRet0 = 150

0.2

0.1

0

−0.1

−0.2

0.2

0.1

0

−0.1

0.3

0.2

0.1

0

0 0.1 0.2y/d

0.3 0.4

−0.1

LOS

SG

AIN

LOS

SG

AIN

LOS

SG

AIN

Figure 8.13. Budgets of turbulent kinetic energy.

the redistribution from the v2 to u2 components. As for Case 3, the effects from thetwo-dimensional vortex and the reattachment are combined and they affect the redis-tribution mechanism. As a result, as shown in Figure 8.12, in spite of the existingrough wall, the anisotropy of turbulence shows behavior similar to that in the smoothwall, and the roughness promotes the heat transfer with a relatively small drag.

Finally, Figure 8.15 shows the streamwise vortices educed with the secondinvariant tensors of the velocity gradient II [36, 37]. In each case, streamwisevortices are produced in the region above the ribs on the rough-wall side, andthe structures spread over the region above the center of the channel. Moreover,



Case 1 H = 0.2δ

Case 3 H = 0.05δ

Inside of rib

0.1

0

−0.1

Φ+ ij

0.1

0

−0.1

Φ+ ij

Φ11Φ22Φ33

20 40 60 800

y+

Symbols: plane channel

Figure 8.14. Distributions of pressure strain terms in transport equations forReynolds stress.

if we compare Case 1 with Case 3, many more streamwise vortices are producedin Case 1, where the rib effects are strong. In the enclosure between the ribs,the streamwise vortices are seen in the whole region of the enclosure in Case 3.However, in Case 1, because the flow stagnates downstream of the rib surface,not many streamwise vortices are produced. This corresponds to the results ofthe distributions of the skin friction coefficient and the Nusselt number as seen inFigures 8.7 and 8.8.

8.5.4 Statistical characteristics of thermal field and related turbulentstructures

For the thermal field, the turbulence statistics at the middle of the cross-sectionalenclosure (x/δ= 0) are discussed below. Figure 8.16 shows the mean temperaturedistribution normalized by the temperature difference Tw. The effect of the ribcauses the mean temperature to become asymmetric in all cases as for the meanvelocity profiles, and the temperature increases in the major part of the channel.On the other hand, because of the rough-wall side effects, the region of the largetemperature gradient extends to the smooth-wall side. Figure 8.17 shows the tur-bulent heat flux in the wall-normal direction. The heat transfer is activated from



x

y

x Case 1 H = 2δ

Case 3 H = 0.05δ

y

Figure 8.15. Vortex structures visualized by second invariant: II∗<−0.5.

0.3

0.2

0.1

0.1 10 20 0.2

0.2

0.4

0.6

0.8

Plane channel

Ret 0 = 150

Pr = 0.71

y /δ

1Rough wall Smooth wall


Θ/∆

Tw

Figure 8.16. Profiles of mean temperature.

the enclosure with the ribs, but in the thermal field also there are counter gradientdiffusions in Case 1.

Next, Figure 8.18 shows the rms intensities of temperature fluctuation, whichdecrease on the rough-wall side, where the velocity fluctuations are promoted.However, the temperature intensities increase on the smooth-wall side, wherethe turbulent velocity fluctuation is suppressed. Apparently, the contrary situa-tions occur between the velocity and thermal fields on each side. To investigatethe cause and effect, we calculated the production and turbulent diffusion termsof the transport equation for θ2/2 see equation (11) in each case (Figure 8.19).




Plane channel

10

0−vθ

/Um

∆Tw

0.002

2

Smooth wall

y/δ

Rough wall

Figure 8.17. Profiles of wall-normal turbulent heat flux.

Case 1 H = 0.2δCase 2 H = 0.1δCase 3 H = 0.05δPlane channel

Smooth wall

y/δ1 20

0.05

θ rm

s/∆T

w

0.1

Rough wall

Figure 8.18. Profiles of rms intensities of temperature fluctuation.

The production terms on the smooth-wall side, where the temperature gradient islarge and the wall-normal heat fluxes increase, contributes greatly in the regionaround the peak to the center of the channel. This is why the temperature intensi-ties become large on the smooth-wall side. On the other hand, on the rough-wallside, because the contribution from the production decreases and the turbulenceis maintained by the turbulent diffusion, the intensities of temperature fluctuationsdecrease, especially in Case 1.

In order to examine the relationship between the velocity and thermal fields,the distributions of the turbulent Prandtl number, Prt, are shown in Figure 8.20.The spanwise averaged Prt at the middle of the enclosure and of the rib crestare compared with that in the plane channel flow. Near the region above the rib,Prt becomes larger than that on the rib crest. However, it becomes smaller andthen becomes larger again in the enclosure. Thus, the turbulent Prandtl numberis not constant and the analogy between heat and momentum transfer cannot beexpected near the rib roughness. On the other hand, on the smooth-wall side, thePrt distribution well corresponds to that in the plane channel flow.



0.02Case 1 H = 0.2δCase 2 H = 0.1δCase 3 H = 0.05δPlane channel0.01

Production

Turbulent diffusion

00.005

0

−0.005

0 1y/δ

2

Figure 8.19. Production and turbulent diffusion terms in temperature variancebudget.

2

1

0

Pr t

y/ δ

1 2

Spanwise average at themiddle of enclosure

Spanwise average at themiddle of rib crest

Plane channel

Case 1 H = 0.2δ

Rough wall Smooth wallRib

Ret0 = 150

Figure 8.20. Turbulent Prandtl number.

Finally, to observe the spatial structures that contribute to the transport of thepassive scalar, Figure 8.21 shows the streamlines, which are spatially averagedalong the streamwise direction, and the temperature fluctuations in the y–z crosssection in the plane channel flow and Case 1. In the plane channel flow, the stream-lines show the streamwise vortices in the near-wall region; the sharp variationsin the temperature fluctuations associate with the vortices. On the other hand, inCase 1, large-scale vortex structures appear, which extend to the center of the chan-nel from the enclosure between the ribs. With this vortex structure, the turbulentmixing becomes larger in the center of the channel, and the turbulence transport ispromoted. This behavior is consistent with the above-mentioned statistical results,i.e., the increase in the heat transfer coefficient, the pronounced change in the mean



Plane channel Cooled wall

Heated wall

Smooth wall (cooled)

Rough wall (heated)

Rib

Case 1 H = 0.2d

y

z

Figure 8.21. Streamwise-averaged streamlines and temperature fluctuations in y–zplane: θ/Tw = −0.05 (white) 0.05 (black).

temperature distributions, and the increase in the turbulent diffusion terms in thebudget of temperature variance.

8.6 DNS of Turbulent heat Transfer in Channel Flow withArbitrary Rotating Axes: Spectral Method

In order to determine improvements in the modeled expression of rotating, wall-bounded turbulent shear flows, DNSs of fully developed channel flows with stream-wise, wall-normal, and spanwise rotation are carried out using the spectral methodfor the 15 cases indicated inTable 8.5 [18]. DNS results of cases forWNR1∼WNR5and STR1∼STR5 are shown in Figures 8.22 and 8.23, respectively (Case 3 is notshown here). In these cases, the spanwise mean velocity appears to be caused by arotational effect, which increases with the increase in rotation number in both cases.In particular, with the increase in the spanwise mean velocity of Case 1, streamwisemean velocity decreases due to the exchange momentum between the streamwiseand spanwise velocitites as in the following equations obtained from equation (5):

0 = − 1

ρ

∂P

∂x+ ∂

∂y

(ν∂U

∂y− uv

)− 2W (17)

0 = ∂

∂y

(ν∂W

∂y− vw

)+ 2U (18)



Table 8.5. Computational conditions

Reτ 150

Pr 0.71

Case 1: Wall-normal rotation

WNR1 WNR2 WNR3 WNR4 WNR5

Roτ2 0.01 0.02 0.05 0.1 0.3

Case 2: Streamwise rotation

STR1 STR2 STR3 STR4 STR5

Roτ1 1.0 2.5 7.5 10.0 15.0

Case 3: Spanwise rotation

SPR1 SPR2 SPR3 SPR4 SPR5

Roτ3 0.0 0.75 1.5 3.0 5.0

Therefore, to predict the flow of Case 1 exactly, the spanwise mean velocityshould be reproduced by a turbulence model. Note that the Reynolds shear stress,uv, tends to decrease remarkably with the increase in rotational number as shownin Figure 8.22b.

On the other hand, it is found from DNS that cases of streamwise rotatingchannel flow (Case 2) involve the counter gradient turbulent diffusion in the span-wise direction shown in Figure 8.23. In view of turbulence modeling, this factclearly demonstrates that the linear “eddy diffusivity turbulence model” (EDM)and the quadratic “nonlinear eddy diffusivity turbulence model” (NLEDM) can-not be applied to calculate the case of a streamwise rotating channel flow. Thus,the following modeled equation 19 employed in the linear EDM and the quadraticNLEDM cannot clearly express a counter gradient turbulent diffusion:

−vw = νt∂W

∂y(19)

where the Reynolds shear stresses of a quadratic NLEDM are expressed identical toa linear model. Also, noted that the rotational term does not appear in the momen-tum equation of a fully developed streamwise rotating channel flow indicated asequation (22), in which the rotational effect is included implicitly in the Reynoldsshear stress in the same manner as the spanwise rotational flow [38]. Therefore, acubic NLEDM or Reynolds stress equation model (RSM) should be used for thecalculation of streamwise rotating channel flows as follows:

0 = − 1

ρ

∂P

∂x+ ∂

∂y

(ν∂U

∂y− uv

)(20)



(a) (b)

(c) (d)

(e)

20

y/δ

U+

w+

uw

+u

v+

100

−1

−0.4

−0.2

1

0 1 2

y/δ0 1 2

y/δ0

10

5

15

0.2

0

0.4

3

4

1

2

1 2

y/δ0 1 2

y/δ0 1 2

Roτ = 0.01 Roτ = 0.10Roτ = 0.15

Roτ = 0.10Roτ = 0.15

Roτ = 0.02Roτ = 0.04

Roτ = 0.01 Roτ = 0.10Roτ = 0.15Roτ = 0.02

Roτ = 0.04

Roτ = 0.01Roτ = 0.02Roτ = 0.04

Roτ = 0.10Roτ = 0.15

Roτ = 0.01Roτ = 0.02Roτ = 0.04

Roτ = 0.10Roτ = 0.15

Roτ = 0.01Roτ = 0.02Roτ = 0.04

k

Figure 8.22. DNS results of wall-normal rotating flows (Case 1): (a) streamwisemean velocities, (b) Reynolds shear stresses, uv, (c) spanwise meanvelocities, (d) Reynolds shear stresses, vw, and (e) turbulent kineticenergy.

0 = − 1

ρ

∂P

∂y− ∂v2

∂y+ 2W (21)

0 = ∂

∂y

(ν∂W

∂y− vw

)(22)

As previously stated, one of the main objectives in conducting DNS is to obtainthe detailed data for the evaluation of the turbulence models. Thus, this section



20

10

0 1

(a)

2

1

0

0.3

0.2

0.1

00 1

(d)

0 1

(b)

2−1

1

0

−1

U+

W+

k+

uv+

uw+

Roτ=1.0Roτ=2.5Roτ=7.5Roτ=10Roτ=15





y /δ

0 1

(c)

1

2

3

0 1

(e)

2

4

2 2y /δ

y /δ

y /δ

y /δ

Figure 8.23. DNS results of streamwise rotating flows (Case 2): (a) streamwisemean velocities, (b) Reynolds shear stresses, uv, (c) spanwise meanvelocities, (d) Reynolds shear stresses, vw, and (e) turbulent kineticenergy.

briefly explains the characteristics of channel flows with heat transfer, and withwall-normal or streamwise rotation [39]. DNSs of fully developed nonisothermalchannel flows with wall-normal (Case 1) or streamwise (Case 2) rotation are carriedout for the 10 cases indicated in Table 8.5. DNS results are shown in Figures 8.24–8.27. In these cases, the spanwise mean velocity appears to be caused by a rotationaleffect, which increases with the increase in rotation number in both cases. Also,Reynolds shear stress, vw, and the spanwise turbulent heat flux, wθ, are yieldedin both velocity and thermal fields as indicated in Figures 8.25 and 8.27 (note that



0.8

50

40

30

20

10

0

0.6

0.4

0.2

0 1 2 1 2y/δ y/δ

Roτ = 0.01Roτ = 0.02

Roτ = 0.15Roτ = 0.10Roτ = 0.04

Roτ = 0.01Roτ = 0.02

Roτ = 0.15Roτ = 0.10Roτ = 0.04

kθ+

(Θ −

ΘH)/

∆Θ

Figure 8.24. Distributions of mean temperature and temperature fluctuation kθ(Case 1).

(a) (b)

(c)

ROτ = 0.01 ROτ = 0.10ROτ = 0.15

−5

5

0

−5

5

0

0 1 2

0.5vθ+

vθ+

1

vθ+

y/δ

0 1 2y/δ

0 1 2y/δ

ROτ = 0.02ROτ = 0.04

ROτ = 0.01ROτ = 0.02ROτ = 0.04ROτ = 0.10ROτ = 0.15

ROτ = 0.01ROτ = 0.02ROτ = 0.04ROτ = 0.10ROτ = 0.15

Figure 8.25. Turbulent heat fluxes (Case 1): (a) streamwise, (b) wall-normal,(c) spanwise.

turbulent quantities of velocity field are omitted here; for details, see Ref. [18]). It iswell known that Reynolds shear stress, vw, remarkably relates to the distribution ofspanwise mean velocity, but the spanwise turbulent heat flux, wθ, hardly affects thedistribution of mean temperature due to the absence of mean temperature gradientin the spanwise direction.

In Case 1, it can be seen that the profiles of mean temperature scarcely vary,but the temperature fluctuations increase distinctly with the increase in rotationnumber. As for the turbulent heat fluxes, decrease of the streamwise turbulent heat



0 1y/δ

2

0.2

0.4

0.6

0.8

1

Roτ = 10Roτ = 15

Roτ = 1.0Roτ = 2.5Roτ = 7.5

0 1y/δ

2

1

2

3

4

5

6

Roτ = 10Roτ = 15

Roτ = 1.0Roτ = 2.5Roτ = 7.5

k θ+

(Θ −

ΘH)∆

Θ

Figure 8.26. Distributions of mean temperature and temperature fluctuation kθ(Case 2).

Roτ = 10Roτ = 15

Roτ = 1.0Roτ = 2.5Roτ = 7.5

1

0.5

1.5

0 1y/δ

2

(c)

0.5

0 1 2

5

y/δ

Roτ = 10Roτ = 15

Roτ = 1.0Roτ = 2.5Roτ = 7.5

0−5

0

5

1y/δ

2

Roτ = 10Roτ = 15

Roτ = 1.0Roτ = 2.5Roτ = 7.5

uu

+

uu

+

uu

+

(a) (b)

Figure 8.27. Turbulent heat fluxes (Case 2): (a) streamwise, (b) wall-normal,(c) spanwise.

flux, and increase in the spanwise turbulent heat flux with large rotation number areobserved as shown in Figure 8.25. The streamwise turbulent heat flux ultimatelyindicates an inverse sign at the largest rotation number. On the other hand, the wall-normal turbulent heat flux, which affects mean temperature distribution is kept inthe region of the channel center, while the decrease is found near the wall with theincrease in rotation number.

As for Case 2, since the streamwise mean velocity decreases little and thespanwise mean velocity fluctuates significantly and increases with the increase in



rotation number [18], remarkable increases in the spanwise turbulent heat flux, wθ,are also observed near the wall and center of the channel. However, the mean tem-perature and the other turbulent heat fluxes of this case are hardly affected by therotation as indicated in Figures 8.26 and 8.27. Note that the counter-gradient dif-fusion phenomenon was found between the spanwise mean velocity and Reynoldsshear stress, vw in Case 2 [18]. Therefore, the linear eddy diffusivity model cannotaccurately predict in this case [18].

In view of turbulence modeling, this fact clearly demonstrates that the lineareddy diffusivity models cannot be applied to calculate the thermal field of rotatingchannel flow, because it is difficult to include the rotational effect in the linear eddydiffusivity model. On the other hand, it is well known that it is difficult for a lineareddy diffusivity model to predict the streamwise turbulent heat flux. Although thespanwise heat flux also appears in these cases, the model might not predict it byreason of the modeled hypothesis, i.e., gradient diffusion modeling. Therefore, inorder to accurately predict the heat transfer of rotating channel flows, the turbulentheat-flux equation model, the algebraic turbulent heat-flux model, or the nonlineareddy diffusivity model should be used. In this study, the nonlinear eddy diffusivityfor heat model (NLEDHM) [16] is developed to predict adequately these heattransfer phenomena of rotating channel flows.

8.7 Nonlinear Eddy Diffusivity Model for Wall-BoundedTurbulent Flow

8.7.1 Evaluations of existing turbulence models in rotatingwall-bounded flows

The evaluated models are a cubic model by Craft et al. [40] (hereinafter referred toas NLCLS) and a quadratic model by Nagano and Hattori [38] (NLHN).

Evaluations of wall-normal rotating flows predicted by the NLCLS and theNLHN models are indicated in Figure 8.28. In this case, it can be seen thatthe NLHN accurately reproduces turbulent quantities. Thus, one may considerthat streamwise rotating flows can be predicted using a quadratic nonlinearmodel. However, the wall-limiting behavior of the Reynolds stress componentin the wall-normal direction is not predicted in the vicinity of the wall shown inFigure 8.28f.

Figure 8.29 shows the assessment result using the present DNS of two-equationmodels with NLEDM in the case of streamwise rotating channel flow. Obviously, thequadratic model cannot reproduce this flow as mentioned in the previous section.The cubic model indicates overprediction of mean streamwise velocity, U , andunderpredicts the Reynolds shear stress, vw, in the case of a higher rotation number.The wall-limiting behavior of Reynolds stress components is shown in Figure 8.29f.Although the NLHN model is modeled to satisfy the wall-limiting behaviour of theReynolds stress component in the wall-normal direction in spanwise rotational flow,it can be seen that the model does not reproduce the wall-limiting behavior.



Roτ = 0.15

Roτ = 0.10

Roτ = 0.04

Roτ = 0.02

Roτ = 0.01

DNS

10

0

U+

0

0

0

0 1

(a)

2y/δ

NLCLSNLHN

Roτ = 0.15

Roτ = 0.10

Roτ = 0.04

Roτ = 0.02

Roτ = 0.01

20

10

W+

0

0

0

0

0 1 2

(c)y/δ

DNSNLCLSNLHN

Roτ = 0.15

Roτ = 0.10

Roτ = 0.04

Roτ = 0.02

Roτ = 0.01

5

0

K+

0

0

0

0

(f)

0 1 2

y/δ

DNSNLCLSNLHN

Roτ = 0.15

Roτ = 0.10

Roτ = 0.04

Roτ = 0.02

Roτ = 0.01

uv+

0.5

0

0

0

0

0

−0.5

(d)

0 1 2

y/δ

DNSNLCLSNLHN

Roτ = 0.15

Roτ = 0.10

Roτ = 0.04

Roτ = 0.02

Roτ = 0.01

1

0

uv+

0

0

0

0

−1

(b)

0 1 2

y/δ

DNSNLCLSNLHN

Lines: NLHN

u 2+

v 2+

w 2+

Roτ = 0.1

(g)

Symbols: DNS

10−2

10−0

10−2

10−4

10−6

10−8

10−10

10−12

10−14

10−1 100 101

y +102

u2+

v2+

w2+

,,

Figure 8.28. Evaluations of predicted wall-normal rotating flows (Case 1):(a) streamwise mean velocities, (b) Reynolds shear stresses, uv,(c) spanwise mean velocities, (d) Reynolds shear stresses, vw, (e) tur-bulent kinetic energy, and (f) wall-limiting behavior of Reynolds stresscomponents.



Roτ = 15

Roτ = 7.5

Roτ = 2.5

Roτ = 1.0

30

20

10

0

0

0

0

0

U+

DNSNLCLSNLHN

y/δ1 2

Roτ = 10

(a)

Roτ = 15

Roτ = 10

Roτ = 7.5

Roτ = 2.5

Roτ = 1.0

0

0

0

0

0

uv+

DNSNLCLSNLHN

y/δ

0 1 2

1

−1

(b)

y/δ

(c)

y/δ

(d)

y/δ

(e)

(f)

W+

k+

DNSNLCLSNLHN

Roτ = 15

Roτ = 7.5

Roτ = 2.5

Roτ = 1.0

Roτ = 10

1.5

0

0

0

0

0

−0.510 2

DNSNLCLSNLHN

Roτ = 15

Roτ = 7.5

Roτ = 2.5

Roτ = 1.0

Roτ = 10

0

5

0

0

0

10 2

uw+

DNSNLCLSNLHN

Roτ = 15

Roτ = 7.5

Roτ = 2.5

Roτ = 1.0

Roτ = 100

0.2

0

0

0

0

1 2

y +

100

10−2

10−4

10−6

10−8

10−10

10−2 10−1 10−0 10−1 10−2

u2+

v2+

w2+

,,

Lines: NLHN

u 2+

v 2+

w 2+

Roτ = 0.1

Symbols: DNS

Figure 8.29. Evaluations of predicted streamwise rotating flows (Case 2):(a) streamwise mean velocities, (b) Reynolds shear stresses, uv,(c) spanwise mean velocities, (d) Reynolds shear stresses, vw, (e) tur-bulent kinetic energy, and (f) wall-limiting behavior of Reynolds stresscomponents.



8.7.2 Proposal of nonlinear eddy diffusivity model for wall-bounded flow

From these results, we propose a cubic NLEDM in a two-equation turbulence modelwhich can adequately predict rotational channel flows with arbitrary rotating axes,in which a modeling of wall-limiting behavior of Reynolds stress components isalso considered.

The transport equation of Reynolds stress with the Coriolis term is given asfollows:

Duiuj

Dt= Dij + Tij + Pij + Cij +ij − εij (23)

where Dij is a molecular diffusion term, Tij is a turbulent and pressure diffusionterm, Pij = −uiuk (∂U j/∂xk ) − ujuk (∂U i/∂xk ) is a production term, Cij = −2m(εmkjuiuk + εmkiujuk ) is a Coriolis term, ij is a pressure–stain correlation termand εij is a dissipation term, respectively.

Introducing the Reynolds stress anisotropy tensor bij = uiuj/2k − δij/3 andneglecting the diffusive effect, the following relation is derived from equations (9)and (23):

Dbij

Dt= 1

2k(Pij + Cij +ij − εij) − bij + δij/3

k(Pk − ε) (24)

In the local equilibrium state, since the relation Dbij/Dt = 0 holds, equation (24)yields the following relation:

(Pij + Cij +ij − εij) = 2(

bij + δij

3

)(Pk − ε) (25)

Using the form εij = 23εδij + Dεij of the dissipation term [41] for equation (25),

we can obtain:

(Pk − ε)bij = − 2

3kSij − k

(bik Sjk + bjk Sik − 2

3bmnSmnδij

)(26)

− k[bik (Wjk + 2εmkjm) + bjk (Wik + 2εmkim)] + 1

2ij

where ij =ij − Dεij , and the modeled ij is employed as the following generallinear model:

ij = − C1εbij + C2kSij + C3k

(bik Sjk + bjk Sik − 2

3bmnSmnδij

)(27)+ C4k(bik Wjk + bjk Wik )

where C1 ∼ C5 are model constants.



Substituting equation (27) into (26) and introducing nondimensional quantities,we can obtain the following relation:

b∗ij = −S∗

ij −(

b∗ik S∗

jk + b∗jk S∗

ik − 2

3b∗

kS∗kδij

)+ b∗

ik W ∗kj + b∗

jk W ∗ki (28)

where

S∗ij = 1

2gτ(2 − C3)Sij , W ∗

ij = 12

gτ(2 − C4)[ij +

(C4 − 4C4 − 2

)εmjim

]b∗

ij =(

C3 − 2

C2 − 43

)bij, τ = k

ε, g =

(1

2C1 + Pk

ε+ Gk

ε− 1

)−1

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (29)

Equation (28) can be written in the matrix form as:

b∗ = −S∗ −(

b∗S∗ + S∗b∗ − 23b∗S∗I

)+ b∗W∗ − W∗b∗ (30)

In order to derive an explicit form of b∗ from equation (30), the integrity basis,b∗ = ∑

λQ(λ)T(λ) first proposed by Pope [42], is used with the following 10 basistensors:

T(1) = S∗, T(6) = W∗2S∗ + S∗W∗2 − 23 S∗W∗2I

T(2) = S∗W∗ − W∗S∗, T(7) = W∗S∗W∗2 − W∗2S∗W∗

T(3) = S∗2 − 13 S∗2I, T(8) = S∗W∗S∗2 − S∗2W∗S∗

T(4) = W∗2 − 13 W∗2I, T(9) = W∗2S∗2 + S∗2W∗2 − 2

3 S∗2W∗2IT(5) = W∗S∗2 − S∗2W∗, T(10) = W∗S∗2W∗2 − W∗2S∗2W∗

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭(31)

Substituting equation (31) into b∗ = ∑λ Q(λ)T(λ) gives:

b∗ = Q(1)S∗ + Q(2)(S∗W∗ − W∗S∗) + Q(3)(

S∗2 − 13S∗2I

)+ Q(4)

(W∗2 − 1

3W∗2I

)+ Q(5)(W∗S∗2 − S∗2W∗)

+ Q(6)(

W∗2S∗ + S∗W∗2 − 2

3S∗W∗2I

)(32)

+ Q(7)(W∗S∗W∗2 − W∗2S∗W∗) + Q(8)(S∗W∗S∗2 − S∗2W∗S∗)

+ Q(9)(

W∗2S∗2 + S∗2W∗2 − 23S∗2W∗2I

)+ Q(10)(W∗S∗2W∗2 − W∗2S∗2W∗)



On the other hand, substituting the integrity basis and the theoretical expres-sions [42]:

T(λ)S∗ + S∗T(λ)−23T(λ)S∗I =

∑γ

HλγT(γ)

and

T(λ)W∗ − W∗T(λ) =∑γ

JλγT(γ)

where H and J are the scalar functions, into equation 30, gives the followingequation for T(λ):

∑λ

Q(λ)T(λ) = −∑λ

δ1λT(1) −∑λ

Q(λ)

[(∑λ

HλγT(λ)

)−

(∑λ

JλγT(λ)

)](33)

Equation (33) can be written as Q(λ) = A−1γλ Bλ, where Aγλ= −δλγ − Hλγ + Jλγ ,

and Bλ= δ1λ. By using Cayley–Hamilton identities, the matrices Hλγ and Jλγ canbe determined. Thus, we can obtain Q(λ) using Mathematica as follows:

Q(1) = − 12 (6 − 3η1 − 21η2 − 2η3 + 30η4)/D Q(6) = −9/D

Q(2) = −(3 + 3η1 − 6η2 + 2η3 + 6η4)/D Q(7) = 9/D

Q(3) = (6 − 3η1 − 12η2 − 2η3 − 6η4)/D Q(8) = 9/D

Q(4) = −3(3η1 + 2η3 + 6η4)/D Q(9) = 18D

Q(5) = −9/D Q(10) = 0

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭(34)

where

D = 3 − 7

2+ η2

1 − 15

2η2 − 8η1η2 + 3η2

2 − η3 + 2

3η1η3

(35)− 2η2η3 + 21η4 + 24η5 + 2η1η4 − 6η2η4

with η1 = S∗2, η2 = W∗2, η3 = S∗3, η4 = S∗W∗2, and η5 = S∗2W∗2.In order to obtain reasonable forms for Q(1) ∼ Q(9), we assume S13 = W13 = 0.

Thus, η3 and η4 become 0 and η5 = 12η1η2. Therefore, the following reasonable

forms for Q(1) ∼ Q(6) can be derived:

Q(1) = − 3

3 − 2η1 − 6η2, Q(2) = − 3

3 − 2η1 − 6η2, Q(3) = 6

3 − 2η1 − 6η2

Q(4) = −18η1

(3 − 2η1 − 6η2)(2 − η1 − η2), Q(5) = −18

(3 − 2η1 − 6η2)(2 − η1 − η2)

Q(6) = −18(3 − 2η1 − 6η2)(2 − η1 − η2)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭(36)



Consequently, we can obtain the cubic NLEDM as follows:

b∗ij = − 3

3 − 2η2 + 6ζ2

S∗

ij + (S∗ik W ∗

kj − W ∗ik S∗

kj) − 2(

S∗ik S∗

kj − 1

3S∗

mnS∗nmδij

)+ 6

2 − η2 + ζ2

[η2(W ∗

ik W ∗kj − 1

3δijW

∗mnW ∗

nm) + (W ∗ik S∗

kS∗j − S∗

ik S∗kW

∗j)

+(

W ∗ikW ∗

kS∗j + S∗

ik W ∗kW

∗j − 2

3S∗mW ∗

mnWnδij

)](37)

where η= (S∗ijS

∗ij)

12 , ζ= (W ∗

ij W ∗ij )

12 , and b∗

ij, S∗ij , and W ∗

ij are nondimensionalquantities, respectively, redefined as follows:

b∗ij = CDbij, S∗

ij = CDτRo Sij, W ∗ij = 2CDτRo Wij (38)

Since the proposed model in equation (37) can be written as the following equa-tion for the Reynolds shear stress, vw (or b23), in streamwise rotating flow, in whichS12(= S21), W12(= −W21), S23(= S32), and W23(= −W32) exist, the Reynolds shearstress, vw, can be reproduced by the present model:

b∗23 = 3

3 − 2η2 + 6ζ2

S∗

23 + 6

2 − η2 + ζ2 [(W ∗21S∗

12S∗23 − S∗

21S∗12W ∗

23)(39)

+ (W ∗21W ∗

12S∗23 + W ∗

23W ∗32S∗

23 + S∗21W ∗

12W ∗23 + S∗

23W ∗32W ∗

23)]

In equation (37), the functions 3/(3 − 2η− 2 + 6ζ2) and 6/(2 − η2 + ζ2) maybe taken to be a negative value or 0 with the increase in η. Thus, in order to avoidtaking a negative value or 0, we model these functions taking into account therotational effect as follows:

3

3 − 2η2 + 6ζ2 1[1 + 22

3

(W ∗2

4

)+ 2

3

(W ∗2

4− S∗2 − f ∗2

ω

)fB + 2

3f ∗2ω

](40)

62 − η2 + ζ2 3[

1 + 3

2

(W ∗2

4

)+ 1

2

(W ∗2

4− S∗2 − f ∗2

ω

)fB + 1

2f ∗2ω

] (41)

where

fB = 1 + Cη

(W ∗2

4− S∗2 − f ∗2

ω

)(42)

f ∗2ω = (CDτRo )

2[m(2εmjiWij −m)]2 (43)



The model function, f ∗2ω , in equation (43) is introduced to avoid an inappropriate

value of (W ∗2/4 − S∗2) with the increase in a rotation number. Since f ∗2ω = 0 is

consistently kept in non rotational flows, inadequate Reynolds stresses predictedby the proposed model are not given in non rotational flows.

Finally, in order to construct a precise model for rotational flow, a cubic termintroduced in an NLCLS model [40] is added in the proposed model. Consequently,the final form of the present cubic NLEDM is given as follows:

uiuj = 2

3kδij − c1νtSij + c2k(τ2

Ro+ τ2

Rw)(Sik Wkj − Wik Skj)

+ c3k(τ2Ro

+ τ2Rw

)(

Sik Skj − 1

3SmnSmnδij

)+ c4kτ2

Ro

(Wik Wkj − 1

3WmnWmnδij

)(44)

+ c5kτ3Ro(Wik SkSj − SikSkWj) + c6kτ3

Ro

(Wik WkSj + Sik WkWj

− 2

3SmWmnWnδij

)+ c7kτ3

Ro(SijSkSk − SijWkWk)

where νt = Cµfµk2/ε and Cµ= 0.12, and the model functions and constants aregiven as follows:

c1 = −2/fR1, c2 = −4CD/fR1, c3 = 4CD/fR1

c4 = −8C3DS2/( fR1 fR2), c5 = 4C2

D fR/( fR1 fR2),

c6 = 2C2D/( fR1 fR2), c7 = 10C2

µC2D, CD = 0.8

fR1 = 1 + (CDτRo)2 [(22/3)W 2 + (2/3) (W 2 − S2 − f 2ω ) fB]

fR2 = 1 + (CDτRo)2 [(3/2)W 2 + (1/2) (W 2 − S2 − f 2ω ) fB]

fR = 12 [1 − exp ( f 3/4SW /26)]

fµ = [1 − fw(32)] 1 + (40/R3/4t ) exp[−(Rtm/35)3/4]

fB = 1 + Cη (CDτRo)2 (W 2 − S2 − f 2ω )

f 2ω = m(2εmjiWij −m), Cη = 5.0

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(45)

The characteristic time-scale τRw is introduced in an NLHN model [38] for ananisotropy and wall-limiting behavior of turbulent intensity defined as:

τRw =√

16

fR1/CD

fSW

(1 − 3Cv1fv2

8

)f 2v1 (46)



Table 8.6. Model constants and functions of transport equations for k and ε

Cε1 Cε2 Cε3 Cε4 Cε5 Cs Cε C Cf

1.45 1.9 0.02 0.5 0.015 1.4 1.4 −0.045 6.0

ft1 ft2 fε

1 + 9fw(8)

[1 − fw(32)]1/2

1 + 5fw(8)

[1 − fw(32)]1/2

1 − 0.3 exp

[−

(Rt

6.5

)2]

[1 − f 2w (3.7)]

f R

Cf exp

[−

(R10

)0.2] √

ν

ε

√f SW

fSW = W 2

2+ S2

3− f SW (47)

f SW =⎡⎣⎛⎝√

S2

2−

√W 2

2

⎞⎠ fw(1)

⎤⎦2

(48)

where Cv1 = 0.4, Cv2 = 2.0 × 103, fv1 = exp [−(Rtm/45)2] and fv2 = 1 −exp (−√

Rt/Cv2).The wall reflection function is defined by fw(ξ) = exp [−(Rtm/ξ)2],and the corrected Reynolds number for the rotating flow is given asRtm = (Ctmn∗R1/4

t )/(CtmR1/4t + n∗) with Ctm = 1.3 × 102 [38]. Moreover, it is found

from the evaluation shown in Figures 8.28f and 8.29f that the wall-limiting behaviorof Reynolds stress component is not satisfied for the tested rotating flows. Thus, themodel function in equation (48) of characteristic time-scale τRw is slightly modifiedas follows:

f SW =∣∣∣∣∣∣√

S∗2

2−

√W ∗2

2

∣∣∣∣∣∣ fw(1)2 (49)

where S∗ = εmnSmn, W ∗ = εmnWmn.For the modeled transport equations of k and ε as indicated in equations (9)

and (10), the turbulent diffusion and the pressure diffusion terms are modeled asfollows [38]:

Tk = ∂

∂xj

(Csft1

νt

kuju

∂k

∂x

)(50)

Tε = ∂

∂xj

(Cεft2

νt

kuju

∂ε

∂x

)(51)



k = max−0.5ν

∂

∂xj

[k

ε

∂ε

∂xjfw(1)

], 0

(52)

ε = Cε4∂

∂xj

[1 − fw(5)]

ε

k

∂ε

∂xjfw(5)

(53)

The extra term in the ε-equation is adopted as the identical to the NLHNmodel [38]:

E = Cε3νk

εuju

∂2U i

∂x∂xk

∂2U j

∂x∂xk+ Cε5ν

k

ε

∂ujuk

∂xj

∂U i

∂xk

∂2U i

∂xj∂xk(54)

Finally, a rotation-influenced addition term, R, in the ε-equation (10) isgeneralized as follows:

R = CfkεijWijd (55)

where d is the unit vector in the spanwise direction. Model constants and functionsin the k- and ε-equations are indicated in Table 8.6. In what follows, we call thecubic NLEDM, thus obtained, a nonlinear eddy diffusivity for momentum model(NLEDMM).

8.8 Nonlinear Eddy Diffusivity Model for Wall-BoundedTurbulent Heat Transfer

8.8.1 Evaluations of turbulent heat transfer models in rotatingchannel flows

To improve the nonlinear eddy diffusivity for heat model (NLEDHM), the followingnonlinear turbulence models are evaluated using DNS data (refer to the detailedmodel functions and constants of each models in Refs. [38], [43], and [44]).

• Nagano and Hattori [38] (hereinafter referred to as NLHN)

ujθ = −αtjk∂

∂xk+ Cθ1

fRTτ2

mouuk(Cθ2Sj + Cθ3Wj + 2Cθ1εjmm

) ∂ ∂xk

(56)

where αtjk is an anisotropic eddy diffusivity tensor for heat as follows:

αtjk =

(C∗θ1

fRT

)ujukτm (57)

• Suga and Abe [43] (hereinafter referred to as NLSA)

uiθ = −Cθkτ(σij + αij)∂

∂xj(58)



0 1

y/δ2

−6

0

0

0

0

Roτ =0.1

Roτ =0.1

Roτ =0.04

Roτ =0.04

Roτ =0.02Roτ =0.02

Roτ =0.01 Roτ =0.01

6DNSNLHN NLYSCNLSA

0 1 2

0

0

0

0

1

DNS NLHNNLSA NLYSC

(a)

y/δ

(b)

uθ+

υθ+

Figure 8.30. Evaluations of turbulent heat fluxes (Case 1); (a) streamwise (uθ),(b) wall-normal (vθ).

where

σij = Cσ0δij + Cσ1uiuj

k+ Cσ2uiu

uuj

k,

αij = Cα0τWij + Cα1τ

(Wi

uuj

k+ Wj

uiuk

)Note that the original model is described in the non rotational form [43]. In orderto evaluate the NLSA model in the rotational flow, the vorticity tensor is replacedwith the absolute vorticity tensor in the model.

• Younis et al. [44] (hereinafter referred to as NLYSC)

uiθ = −C1k2

ε

∂

∂xi− C2

k

εuiuj

∂

∂xj− C3

k3

ε2

∂U i

∂xj

∂

∂xj (59)

−C4k2

ε2

(uiuk

∂U j

∂xk− ujuk

∂U i

∂xk

)∂

∂xj

Note that the above description is in the original model form [44]. In order toapply to the model evaluation in rotating channel flow, the velocity gradient,∂U i/∂xj, must be replaced with Sij + Wij in the model. However, the evaluationis conducted using the original form, because we should know the adverse effectof using it.

Evaluation results are shown in Figures 8.30 and 8.31. In Case 1, it can be seenthat the NLSA model accurately reproduces the streamwise heat flux at variousrotation numbers, because the NLSA model was improved in order to properly pre-dict the streamwise turbulent heat flux [43]. As for an evaluation for a prediction of



0

DNS DNSNLSA

NLSA

0

0

0

0

0

(a)

y /δ

Roτ =1.0

Roτ =2.5

Roτ =7.5

Roτ =10

Roτ =15

Roτ =1.0

Roτ =2.5

Roτ =7.5

Roτ =10

Roτ =15

1 2 0

(b)

y /δ1 2

−6

6

NLHN NLHNNLYSCNLYSC

uθ+

2

0

0

0

0

υθ+

Figure 8.31. Evaluations of turbulent heat fluxes (Case 2); (a) streamwise (uθ),(b) wall-normal (vθ).

wall-normal heat flux, the NLHN model slightly underpredicts in case of the lowestrotation number, but the NLHN model gives good agreement with DNS data withincrease in rotation number. In Case 2, the NLSA and NLYSC give adequate predic-tions of both turbulent heat fluxes in case of the lowest rotation number. Even thoughthe NLYSC does not employ the rotational form, it is noteworthy that the NLYSCmodel nearly gives equivalent results in comparison with the NLSA model. How-ever, it is obvious that all models cannot predict DNS results with increase in rotationnumber. In particular, the wall-normal turbulent heat flux is not reproduced by eval-uated models in case of the highest rotation number. From this evaluation of Case2, it may be concluded that the mean temperature cannot be adequately predictedusing evaluated models. Thus, the model should be improved to accurately predictheated channel flows with arbitrary rotating axes and various rotation numbers.

8.8.2 Proposal of nonlinear eddy diffusivity model for wall-boundedturbulent heat transfer

The formulation of nonlinear eddy diffusivity for a heat model (NLEDHM) isderived below [39]. The transport equation for turbulent heat flux is written:

Dujθ

Dt= Djθ + Tjθ + Pjθ +jθ − εjθ + Cjθ (60)

where we introduce the nondimensional heat-flux parameter a∗j [45]:

a∗j = ujθ√

k√

kθ(61)

where kθ = θ2/2.



The transport equation for a∗j is derived to neglect the diffusive effects as follows:

Da∗j

Dt= D

Dt

(ujθ√k√

kθ

)= 1√

k√

kθ

Dujθ

Dt− 1

2

ujθ√k√

kθ

(1

k

Dk

Dt+ 1

kθ

DkθDt

)= 1√

k√

kθ

(Pjθ +jθ − εjθ + Cjθ

)(62)

−1

2

ujθ√k√

kθ

[ε

k

(Pk

ε− 1

)+ εθ

kθ

(Pkθ

εθ− 1

)]where the transport equations for k and kθ [see equations (9) and (11)] areemployed.

The most general linear expression is adopted for the modeled pressure–temperature gradient correlation term, jθ , and dissipation term, εjθ:

jθ − εjθ = −C1ujθ

τu+ C2ukθ

∂U j

∂xk+ C3ukθ

∂U k

∂xj(63)

where τu = k/ε.The production and Coriolis terms in equation (60) are given by:

Pjθ = −ujuk∂

∂xk− ukθ

∂U j

∂xk(64)

Cjθ = −2εjmkmukθ (65)

Therefore, the first term on the right-hand side of equation (62) is expanded asfollows:

1√k√

kθ(Pjθ +jθ − εjθ + Cjθ) = 1√

k√

kθ

C1

τu

[−ujθ − 1

C1τuujuk

∂

∂xk(66)

− 1 − C2

C1τuukθ

∂U j

∂xk+ C3

C1τuukθ

∂U k

∂xj− 2

C1τuεjmkmukθ

]

where the model constants in equation (66) are replaced as CT1 = 1/C1, CT2 = (1−C2)/C1, and CT3 = C3/C1. Thus:

1√k√

kθ(Pjθ +jθ − εjθ + Cjθ) = 1√

k√

kθ

1

CT1τu

[−ujθ − CT1τuujuk

∂

∂xk

− (CT2 − CT3) τuukθSjk − (CT2 + CT3) τuukθ

(Wjk + 2CT1

CT2 + CT3εjmkm

)](67)



Consequently, substituting equation (67) for equation (62), the followingequation is obtained:

Da∗j

Dt= 1√

k√

kθ

1

Cθ1τu

[−ujθ − Cθ1τuujuk

∂

∂xk− Cθ2τuukθSjk

−Cθ3τuukθ (Wjk + Cθ4εjmkm)]

(68)

− 12

ujθ√k√

kθ

[ε

k

(Pk

ε− 1

)+ εθ

kθ

(Pkθ

εθ− 1

)]where Cθ1 = CT1, Cθ2 = CT2 − CT3, Cθ3 = CT2 + CT3, and Cθ4 = 2CT1/

(CT2 + CT3).In the local equilibrium state, the following relation holds:

Da∗j

Dt= 0 (69)

With the above condition of equation (69), equation (68) becomes as follows:

ujθ√k√

kθ

1

τu

[1 + Cθ1

2

(Pk

ε− 1

)+ Cθ1

2R

(Pkθ

εθ− 1

)](70)

= −(δjk + 3bjk )23

Cθ1k√k√

kθ

∂

∂xk− (Cθ2Sjk + Cθ3W ′

jk )ukθ√k√

kθ

where W ′jk = Wjk + Cθ4εjmkm, and the following characteristic time-scale in the

left-hand side of equation (70) is replaced with the mixed time-scale τm for thenear-wall treatment:

τu

1 + Cθ12

(Pk

ε− 1

)+ Cθ1

2R

(Pkθ

εθ− 1

) → τm (71)

Finally, if we introduce the nondimensional given by the following equations:

b∗jk = 3bjk ,

∗k = 2

3

Cθ1kτm√k√

kθ

∂

∂xk,

S∗jk = Cθ2τmSjk ,

W ∗jk = Cθ3τmWjk .

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭(72)

Thus, equation (70) becomes:

a∗j = −(δjk + b∗

jk ) ∗k − (S∗

jk + W ∗jk )a∗

k (73)



Thus, we obtain the following explicit expression for the nondimensionalturbulent heat flux [45]:

a∗j = 1

1 + 1

2

(W ∗2 − S∗2

) [−δjk − b∗jk + (S∗

jk + W ∗jk ) + (S∗

j + W ∗j) b∗

k ] ∗k (74)

where W ∗2 = W ∗ij W ∗

ij and S∗2 = S∗ijS

∗ij . The derived equation (74) can be rewritten

in the conventional form as follows:

ujθ = − Cθ1τm

fRTujuk

∂

∂xk

+ Cθ1τ2m

fRTuku

(Cθ2Sj

∂

∂xk+ Cθ3Wj

∂

∂xk+ 2Cθ1εjmm

∂

∂xk

)(75)

where Cθ4 = 2Cθ1/Cθ3 and fRT = 1 + 1

2τ2

m(C2θ3W 2 − C2

θ2S2), and the first term of

right-hand side can be rewritten using the following anisotropic eddy diffusivitytensor for heat:

αtjk = Cθ1

fRTujukτm (76)

Thus, the final form for NLEDHM given in equation (56) is obtained asfollows:

ujθ = −αtjk∂

∂xk+ Cθ1

fRTτ2

muku (Cθ2Sj + Cθ3Wj + 2Cθ1εjmm)∂

∂xk(77)

Based on the foregoing, the NLEDHM should be improved to accuratelypredict heated channel flows with arbitrary rotating axes and various rotation num-bers. Although the NLHN properly predicts the rotating heated channel flow witharbitrary rotating axes, the streamwise turbulent heat flux is slightly underpre-dicted. Therefore, in order to construct a turbulence model having higher-predictionprecision, we reconstruct the NLEDHM based on the NLHN model [38].

As for a prediction of streamwise turbulent heat flux, Hishida et al. [46] pro-posed the relation −uθ= Ch(−uv/k)vθ from an experimental result, and Hattoriet al. [24] proposed the modified temperature–pressure gradient correlation anddissipation terms for a buoyancy-affected thermal field, in which the predictionprecision of NLEDHM is improved. In this study, we introduce the dimensionlessReynolds stress tensor Aij(= uiuj/k) in an anisotropic eddy diffusivity tensor forheat as indicated in equation (60), which is adopted in the NLSA model [43] toadequately predict the streamwise turbulent heat flux, for the NLHN model indi-cated in equation [56]. Also, introducing the dimensionless Reynolds stress tensor



into the NLEDHM, the NLEDHM successfully predicts turbulent buoyant flows asmentioned in Hattori et al. [24].

αtjk =

(C∗θ1

fRT

)uiujAikτm (78)

fRT = 1 + 1

2τ2

m [C2θ2(W 2 − S2) + (C2

θ3 − C2θ2)W 2] (79)

where C∗θ1, Cθ2, and Cθ3 are model constants and τm is the characteristic time-scale.

In introducing the dimensionless Reynolds stress tensor Aij, αtjk (∂ /∂xk ) of the

streamwise turbulent heat flux, uθ ( j = 1), is given as follows:

αtxy =

(C∗θ1

fRT

)[uv

k

(u2 + v2

)]τm∂

∂y(80)

where, regarding the derivation of equation (80), a fully developed channel flow isassumed.

The original model of equation (57) makes for the streamwise turbulent heatflux αt

xy = (C∗θ1/fRT )v2τm. Comparing both equations, the effect of dimensionless

Reynolds stress tensor for the prediction of the streamwise turbulent heat flux isobviously obtained, i.e., the streamwise velocity intensity, u2, which has a strongcorrelation with uθ [47], evidently improves the prediction.

8.9 Model Performances

8.9.1 Prediction of rotating channel flow using NLEDMM

The evaluations for the newly improved NLEDM (or NLEDMM) are shown inFigures 8.32–8.42. In order to calculate the present model, the numerical techniqueused is a finite-volume method [48].

First, cases of wall-normal rotating channel flows are shown in Figures [8.32–8.42]. In this case, the streamwise mean velocity decreases with the increasingrotation number due to the exchange momentum between the streamwise and span-wise velocity as indicated in Figures 8.32a and c. Thus, the spanwise velocityincreases at the lower rotation numbers, but the spanwise velocity reaches max-imum value at a critical rotation number. Then, the spanwise velocity decreasesbecause the streamwise velocity becomes almost zero at higher rotation numbers.In comparison with DNS results, the present NLEDMM gives better predictionsthan the predictions of the NLCLS model. In particular, Reynolds shear stresses,vw, are predicted very well as shown in Figure 8.32d. Figure 8.33 shows relationbetween streamwise and spanwise velocities in this case, in which the laminar solu-tions are included for comparison. Obviously, the flow tends to be laminarized withthe increasing rotation number, and the present model can predict properly this ten-dency. Also, anisotropy in turbulent intensities is reproduced by the present cubic



Roτ =0.15

Roτ =0.04

Roτ =0.02

Roτ =0.01

Roτ =0.1

20

10

0

0

0

0

0

W+

DNS NLCLS(1996)Present model

0 1 2

Roτ =0.15

Roτ =0.04

Roτ =0.02

Roτ =0.01

Roτ =0.1

10

0

0

0

0

0

U+

DNSNLCLS(1996)Present model

0

(a)

y /δ y /δ1 2

Roτ =0.15

Roτ =0.1

Roτ =0.04

Roτ =0

Roτ −0.02

Roτ −0.01

1

0

0

0

0

0

0

−1DNS

NLCLS(1996)Present model

0

(b)

(c)

y /δ y /δ(d)

1 2

uυ+

Roτ =0.15

Roτ =0.04

zτ =0.02

Roτ =0.01

Roτ =0.1


0 1 2

5

0

0

0

0

0

−5

υw+

Figure 8.32. Distributions of predicted wall-normal rotating flows (Case 1):(a) streamwise mean velocities, (b) Reynolds shear stresses, uv,(c) spanwise mean velocities, and (d) Reynolds shear stresses, vw.

NLEDMM as shown in Figure 8.34a. As mentioned above, the NLEDMM can pre-dict a turbulent veolocity field similar in many respects to the NLHN model as shownin Figure 8.28, but the wall-limiting behavior is satisfied exactly by the improvedmodel, as indicated in Figure 8.34b. Therefore, it is obvious that the present cubicNLEDMM gives higher performance for the predictions of rotating channel flowswhile maintaining the performance of the basis model, e.g., NLHN model.

Next, the predictions of fully developed streamwise rotating channel flowscalculated by the present model are shown in Figure 8.35. The results with the



Predictions5

4

3

2

1

0 5 10

Roτ =0.10Roτ =0.15Roτ =0.30

: Laminar solution: DNS

U+

W+

Figure 8.33. Relation of between U and W in wall-normal rotating channel flow.

Roτ =0.10 Roτ =0.1Symbols : DNS

Lines : Present model

u2+

υ2+

w2+

6

4

2u2+

υ2+w

2+,

,

0 1 2

(a)

y /δ

100

10−2

10−4

10−6

10−8

10−10

10−12

10−14

10−2 10−1

y+

y+2

y+4

(b)

100 101 102

u2+

υ2+w

2+,

,

Roτ =0.1DNS NLHN Present

u2+

υ2+

w2+

Figure 8.34. Distributions of predicted wall-normal rotating flows (Case 1): (a) tur-bulent intensities and (b) wall-limiting behavior of Reynolds stresscomponents.



Roτ = 1.0

Roτ = 2.5

Roτ = 7.5

Roτ = 10

Roτ = 15

DNS

20

U+

0

0

0

0

0 1 2y/δ


Roτ = 1.0

Roτ = 2.5

Roτ = 7.5

Roτ = 10

Roτ = 151

uυ+

0

0

0

0

0

−10 1 2

y/δ


(a) (b)

Roτ = 1.0

Roτ = 2.5

Roτ = 7.5

Roτ = 10

Roτ = 15

DNS

2

W+

0

0

0

0

0

0 1 2y/δ


−2

Roτ = 1.0

Roτ = 2.5

Roτ = 7.5

Roτ = 10

Roτ = 150.2

uw+

0

0

0

0

0

0 1 2

y/δ


(c) (d)

(e) (f)

0

2

4

6

8100

10−2

10−4

10−6

10−8

10−10

10−12

10−2 10−1 100 101 1021 2y/δ

u2+

v2+

y +2

y +

y +1w2+

Symbols : DNSLines : Preset model

DNS NLHN Present

Roτ =10.0

Roτ =1.0 Roτ =10.0

u2+

υ2+w

2+,

,

u2+

υ2+w

2+,

,

Figure 8.35. Distributions of predicted streamwise rotating flows (Case 2):(a) streamwise mean velocities, (b) Reynolds shear stresses, uv,(c) spanwise mean velocities, (d) Reynolds shear stresses, vw, (e) tur-bulent intensities and (f) wall-limiting behavior of Reynolds stresscomponents.



Ω3

Ω2

2δΩ1

x3

RoτyRoτx

Roτzθ

x2

x1

Flow

Figure 8.36. Coordinate system and arbitrary rotating axis in rotating channel flow.

Rox = 15, Roz = 2.5

Rox = 11, Roz = 2.5

Rox = 7.5, Roz = 2.5

Rox = 5, Roz = 2.5

Rox = 2.5, Roz = 2.5

20

10

0

0

0

0

00 1 2

y/δ

U+

DNSPresent model

Figure 8.37. Distributions of mean velocities in an arbitrary axis rotating channelflow: Case STSP: streamwise mean velocities.

NLCLS model [40] are also included in the figure for comparison. It can be seen inFigures 8.35a–d that the present NLEDMM gives accurate predictions of both meanvelocities and Reynolds shear stresses in all cases, and the test case of the highestrotational number is adequately reproduced by the NLEDMM. Also, the Reynoldsnormal stresses are indicated in Figures 8.35e and f. Now, the present NLEDMMadequately reproduces redistribution of Reynolds normal stresses, and can pre-dict wall-limiting behaviour exactly. These are because the modified characteristictime-scale τRw given as equation (46) is introduced in the proposed NLEDMM.

To confirm the performance of the NLEDMM model, cases of rotating channelflow with combined rotational axes are calculated as shown in Figure 8.36, inwhich DNS databases of these cases are provided by Wu and Kasagi [17]. Therotation numbers of Case STSP are given as Roτx = 2.5 ∼ 15 and Roτz = 2.5, thoseof Case WNSP1 as Roτy = 0.04 and Roτz = 2.5 ∼ 11 and those of Case WNSP2 asRoτy = 0.01 ∼ 0.04 and Roτz = 2.5, respectively. The predicted mean velocities are



1.5

−1.5

y /δ

0

0

0

0

0

DNSPresent model

0 1 2

Rox =2.5 , Roz =2.5

Rox =5 , Roz =2.5

Rox =7.5 , Roz =2.5

Rox =11 , Roz =2.5

Rox =15 , Roz =2.5

W+

Figure 8.38. Distributions of mean velocities in an arbitrary axis rotating channelflow: Case STSP: spanwise mean velocities.

0 1

DNS

y/δ

Present

2

Roy = 0.04,Roz = 2.5

Roy = 0.04,Roz = 5.0

Roy = 0.04,Roz = 7.5

Roy = 0.04,Roz = 11

0

0

0

10

20

U+

Figure 8.39. Distributions of mean velocities in an arbitrary axis rotating channelflow: Case WNSP1: streamwise mean velocities.

shown in Figures 8.37–8.42 with DNS data [17]. It can be seen that the proposedNLEDMM gives accurate predictions of all cases, since the proper modeling forrotating flows with arbitrary rotating axes is introduced in the present two-equationmodel with the cubic NLEDMM.



0 1

DNS

y/δ

Present

2

Roy = 0.04,Roz = 2.5

Roy = 0.04,Roz = 5.0

Roy = 0.04,Roz = 7.5

Roy = 0.04,Roz = 11

0

0

0

10

20

W+

Figure 8.40. Distributions of mean velocities in an arbitrary axis rotating channelflow: Case WNSP1: spanwise mean velocities.

20

0

10

0

0

01 2

y/δ

DNSPresent model

Roy = 0.04

Roy = 0.02

Roy = 0.01

Roz = 2.5

U+

Figure 8.41. Distributions of mean velocities in an arbitrary axis rotating channelflow: Case WNSP2: streamwise mean velocities.

8.9.2 Prediction of rotating channel-flow heat transfer using NLEDHM

Calculations identical with DNS conditions indicated in Table 8.5 using turbulencemodels are carried out in comparison with DNS results. Again, Figures 8.32 and8.35 indicate predicted streamwise and spanwise mean velocities in both cases,and it is obvious that mean velocities in both cases are adequately predicted by theNLEDMM. Since a thermal field is passive scalar, turbulent quantities of veloc-ity field should be accurately predicted by the turbulence model for the velocityfield. Therefore, the proposed NLEDHM can closely incorporate the NLEDMM.



10

0

5

0

0

01 2

y/δ

DNSPresent model

Roy = 0.04

Roy = 0.02

Roy = 0.01Roz = 2.5

W+

Figure 8.42. Distributions of mean velocities in an arbitrary axis rotating channelflow: Case WNSP2: spanwise mean velocities.

1

0

0

0

0 1 2

y/δ

DNSNLHNPresent

Ro τ = 0.10

Ro τ = 0.04

Ro τ = 0.02

Ro τ = 0.01

(Θ −

ΘH)/

∆Θ

Figure 8.43. Predicted mean temperatures (Case 1).

Note that the following results are given from a calculation of full transport equa-tions given in equations (4)–(12) with equation (44) and the present NLEDHM orthe NLHN model.

First, the distributions of mean temperature, which are estimated by the presentNLEDHM are shown in Figure 8.43. The NLHN model (16) is also included inthe figures for comparison. Accurate prediction of the mean temperature of Case1 can be done with both models. On the other hand, the proposed NLEDHMis apparently improved to predict a streamwise turbulent heat flux as indicatedin Figure 8.44a. Although the streamwise turbulent heat flux does not affect thedistribution of mean temperature directly, the turbulent heat transfer model should



(a) (b)

1

0

0

0

0 1 2

DNSNLHNPresent

Ro τ = 0.10

Ro τ = 0.04

Ro τ = 0.02

Ro τ = 0.01

vu+

y /δ

6

0

0

0

0

0 1 2

DNSNLHNPresent−6

uu+

Ro τ = 0.10

Ro τ = 0.04

Ro τ = 0.02

Ro τ = 0.01

y /δ

(c)

DNSPresent

6

0

0

0

0

−6

ωu+

Ro τ = 0.1

Ro τ = 0.04

Ro τ = 0.02

Ro τ = 0.01

0 1 2y /δ

Figure 8.44. Predicted turbulent heat fluxes (Case 1); (a) streamwise (uθ), (b) wall-normal (vθ), (c) spanwise (wθ).

predict the streamwise turbulent heat flux to apply some turbulent heat transferproblem such as buoyant flow. As for the predicted wall-normal heat flux as shownin Figure 8.44b, a discontinuous line can be faintly seen in the channel center region.This is caused by the predicted wall-normal velocity intensity. Since the wall-normalvelocity intensity is slightly overpredicted in this case by the NLEDMM, the wall-normal turbulent heat flux is remarkably affected by the prediction. The predictedspanwise turbulent heat flux is shown in Figure 8.44c. The present predictions arein good agreement with DNS. Figure 8.45 shows the wall-limiting behavior of bothstreamwise and wall-normal turbulent heat fluxes. Since the heat is carried fromthe wall in the case of the heated wall condition, the wall-limiting behavior shouldbe reproduced to exactly estimate heat transfer rate. It can be seen that the present



DNSNLHNPresent

10−1

10−5

10−7

10−3

101

uu+

vu+

10−1 100 101

y+102

y +2

y +3

Rot = 0.04

u u+ v u

+

Figure 8.45. Wall-limiting behavior of turbulent heat fluxes (Case 1).

1

0

0

0

0

0 1 2

y/δ

DNSNLHNPresent

Ro τ − 15

Ro τ 10

Ro τ 7.5

Ro τ 2.5

Ro τ 1.0

(Θ −

ΘH)/

∆Θ

Figure 8.46. Predictions of mean temperature (Case 2).

NLEDHM appropriately predicts both turbulent heat fluxes. Note that the presentNLEDHM exactly satisfies the difference between turbulent heat fluxes, i.e., thestreamwise turbulent heat flux is proportional to y2, and the wall-normal turbulentheat flux is proportional to y3.

In Case 2, although the mean temperature indicated in Figure 8.46 is slightlyunderpredicted by the NLHN model in the highest rotation number, the proposedmodel can satisfactorily reproduce it. Also, in this case, it can be seen that theproposed NLEDHM adequately predicts both the streamwise and the wall-normalturbulent heat fluxes as shown in Figure 8.47. On the other hand, it is difficult for thepresent model to predict the spanwise turbulent heat flux in Case 2. The spanwise



(a)

6

0

0

0

0

0

0 1

y/δ y/δ

2

DNSNLHNPresent−6

Ro τ 15

Ro τ 10

Ro τ 7.5

Ro τ 2.5

Ro τ 1.0

uθ+

(b)

1

0

0

0

0

0 1 2

DNSNLHNPresent

Ro τ = 15

Ro τ = 10

Ro τ = 7.5

Ro τ = 2.5

Ro τ = 1.0

υθ+

Figure 8.47. Predicted turbulent heat flux (Case 2); (a) streamwise (uθ), (b) wall-normal (vθ).

10−1

10−5

10−7

10−1 100 101

y+102

10−3

101

y+2

y+3

Ro τ = 2.5

DNSNLHNPresent

uθ+ υθ+

υθ+

uθ+

Figure 8.48. Wall-limiting behavior of turbulent heat fluxes (Case 2).

turbulent heat flux, wθ, is expressed by the present model in Case 2 as follows:

wθ = −(

C∗θ1

fRT

)(uw

uv

k+ vw v

2

k

)τmo∂

∂y(81)

+ Cθ1fRTτ2

mov2

[(Cθ2 + Cθ3)

∂W

∂y+ 2Cθ11

]∂

∂y

The wall-limiting behaviors of turbulent heat fluxes are shown in Figure 8.48, inwhich the wall-limiting behaviors are properly satisfied by the present NLEDHM.



On the basis of these evidences, it would not be an overstatement to say that theproposed model can be applied to predict the rotating heated channel flows witharbitrary rotating axes.

8.10 Concluding Remarks

Based on our current works, the recent trends in DNS and turbulence models forvelocity and thermal fields have been overviewed in this chapter. Without dispute,the DNS gives us a possibility to obtain highly accurate information on variousturbulence quantities. Within the foreseeable future, DNS researchers will activelytackle more complex problems, e.g., flows at high Reynolds or Prandtl numbers,buoyancy-driven complex flows, and atmospheric boundary layer flows. Of course,it is true that the power of the present supercomputer system does not offer realisticsolutions to the above problems. However, improvement of computer systems, inparticular, the further development of massive parallel-computing systems willconquer most of the serious difficulties which confront us now. In fact, there areemerging industries which aim for petaflops (PFLOPS) computing. New researchesconcerning exascale supercomputing design are also in progress.

The latest turbulence models for velocity and thermal fields deal with the indi-vidual elemental processes of turbulence with the aid of detailed information givenby DNS. As a result, such models can mimic the DNS with high accuracy. Also,many attempts to construct a universal turbulence model are being made. If a num-ber of DNS databases on complex turbulent flows are constructed in the future, wewill be able to approach the universal turbulence model.

Acknowledgment

This study was partially supported by a Grant-in-Aid for Scientific Research (S),17106003, from the Japan Society for the Promotion of Science (JSPS).

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9 Analytical wall-functions of turbulence forcomplex surface flow phenomena

K. SugaOsaka Prefecture University, Sakai, Japan

Abstract

The recently emerged analytical wall-function (AWF) methods for surface boundaryconditions of turbulent flows are summarised. Since the AWFs integrate transportequations of momentum and scalars over the control volumes adjacent to surfaceswith some simplifications, it is easy to include complex surface-flow physics into thewall-function formula. Hence, the AWF schemes are successful to treat turbulentflow and scalar transport near solid, smooth, rough and permeable walls. HighPrandtl or Schmidt number scalar transport near walls or free surfaces are also wellhandled by the AWFs. This chapter, thus, summarises their rationale, fundamentalequations and procedures with illustrative application results.

Keywords: Analytical wall-function, High Prandtl number, High Schmidt number,Permeable wall, Rough wall, Turbulent flows

9.1 Introduction

Wall functions have been used from the early stage of turbulence CFD. Since numer-ical integration of turbulence equations down to a solid wall requires dense gridnodes to resolve the steep variations of the turbulence quantities, it was virtuallyimpossible for the people in 1970s to perform turbulence CFD of engineering flowseven by the most powerful computer in those days. Moreover, flow physics verynear the wall (inside sublayer), which is required for constructing a more advancedturbulence model, was not well known.

However, according to the development of computers, many studies on lowReynolds number (LRN) turbulence models have been made and such LRN mod-els have replaced wall functions in most numerical studies (Launder [1]). Theemergence of direct numerical simulations (Kim, Moin and Moser [2]) enhancedthis tendency, and so many LRN turbulence models have been proposed sincethen.



Despite the fact that many LRN turbulence models perform satisfactorily,industrial engineers still routinely make use of classical wall-function approachesfor representing near-wall turbulence and heat transfer (e.g., Ahmed and Demoulin[3]). One reason for this is that, even with advances in computing power, theirnear-wall resolution requirements make LRN models prohibitively expensive incomplex three-dimensional industrial heat and fluid flows.

This is easily understandable when one considers flows over rough and/orporous surfaces, which are common in industrial applications. Because one cannothope to resolve the details of tiny elements composing the rough and/or porouswalls, the wall-function approach may be the only practical strategy for such indus-trial applications. Another difficulty of performing LRN models is treating scalarfields of high Prandtl or Schmidt numbers. In such flows, the sublayer thickness ofthe scalar is much thinner than that of the viscous sublayer of the flow. This requiresmuch finer grid nodes for the scalar fields than those for flow fields.

The most part of the ‘standard’ log-law wall-function (LWF) strategies wasproposed in the 1970s with the assumption of semi-logarithmic variations of thenear-wall velocity and temperature (e.g. Launder and Spalding [4]). After theestablishment of the LWF, Chieng and Launder [5] improved the wall functionby allowing for a linear variation of both the shear stress and the turbulent kineticenergy across the near-wall cell. Amano [6] also attempted to improve the wallfunctions. However, all the above attempts were still based on the log-law and itis well known that such a condition does not apply in flows with strong pressuregradients and separation (e.g. Launder [1, 7]).

Thus, several attempts for developing new wall-function approaches were made.Recently, Craft et al. [8] proposed an alternative wall-function strategy. While stillsemi-empirical in nature, their model makes assumptions at a deeper, more gen-eral level than the log-law based schemes. This approach is called the analyticalwall-function (AWF) and integrates simplified mean flow and energy equationsanalytically over the control volumes adjacent to the wall, assuming a near-wallvariation of the turbulent viscosity. The resulting analytical expressions then pro-duce the value of the wall shear stress and other quantities which are required overthe nearwall cell.1

Following this strategy, the present author and his colleagues discussed andproposed extensions of the AWF approach that allow for the effects of fine-grainsurface roughness, porous walls and a wide range of Prandtl and Schmidt numbers[10–13]. This chapter summarises the outlines and the application results2 of thoseextensions of the AWF method for complex surface turbulence.

1 In the context of large eddy simulations, Cabot and Moin [9] discussed a wall model whosebasic idea was similar to that of the AWF, though it used an implicit formula of the wall shearstress.2 All the results presented have been computed by in-house finite volume codes which employthe SIMPLE pressure-correction algorithm [14] with Rhie–Chow interpolation [15] and thethird-order MUSCL-type scheme (e.g. [16]) for convection terms.


AWFs of turbulence for complex surface flow phenomena 333

P

n

w

∆y∆x

sU

e

Wall

Figure 9.1. Near-wall grid arrangement.

9.2 Numerical Implementation of Wall Functions

In this section, we recall the main features of the implementation of the wallfunctions into a numerical code.

A simplified transport equation for φ near walls:

∂

∂x(ρUφ) = ∂

∂y

(∂φ

∂y

)+ Sφ (1)

can be integrated using the finite volume method over the cells illustrated inFigure 9.1 giving∫ n

s

∫ e

w

∂

∂x(ρUφ)dx dy =

∫ n

s

∫ e

w

∂

∂y

(∂φ

∂y

)dx dy +

∫ n

s

∫ e

wSφdx dy (2)

[(ρUφ)e − (ρUφ)w]y =[(

dφ

dy

)n−

(

dφ

dy

)s

]x + Sφxy (3)

where Sφ is the averaged source term over the wall-adjacent cell P. Note that xis the wall-parallel coordinate while y is the wall normal coordinate. (Althoughtwo-dimensional forms are written here, extending them to three dimensions isstraightforward.)

When the wall-parallel component of the momentum equation is considered(φ= U ) the term (dφ/dy)s in equation (3) corresponds to the wall shear stressτw, while in the energy equation it corresponds to the wall heat flux qw. Insteadof calculating these from the standard discretization, they are obtained from thealgebraic wall-function expressions.

In the case of the transport equation for the turbulence energy k in incompress-ible flows, the averaged source term over the wall adjacent cell is written as:

Sk = ρPk − ρε = ρ(Pk − ε) (4)

The terms Pk and ε thus also need to be provided by the wall function. Note thatthe wall function also can provide k and ε at P as the boundary conditions for theirtransport equations.



9.3 Standard Log-Law Wall-Function (LWF)

Before introducing the AWF, the standard wall function approach using the log-lawis surveyed briefly.

The first requirement, when using the standard wall function, is that the nodepoint P is sufficiently remote from the wall for y+ to be much greater than thatof the viscous sublayer. The value, at least, greater than 30 (y+ ≥ 30) is normallyapplied. Then, the fluxes of momentum and thermal fields to the wall are supposedto obey the following logarithmic profiles:

U+ = 1

κln y+ + B (5)

+ = 1

κtln y+ + C (6)

where the constants κ= 0.38 to 0.42 and κt = 0.47 to 0.48. The constant B = 5.0 to5.5 while

C = (3.85Pr1/3 − 1.3)2 + 1/κt ln Pr

for smooth wall cases. The wall shear stress τw and heat flux qw at the (N + 1) stepof the iterative computation are then obtained as:

√τw/ρ

(N+1) =(

UP1κ

ln y+P + B

)(N )

(7)

(qw)N+1 =(ρcpUτ P

1κt

ln y+P + C

)(N )

(8)

In the local equilibrium boundary layers, the eddy viscosity formula gives−uv= νt∂U/∂y, and thus the production of the turbulence kinetic energy k canbe written as:

Pk = −uv∂U

∂y= νt

∂U

∂y

(∂U

∂y

)(9)

Then, since νt = cµk2/ε and Pk = ε in local equilibrium,3(−uv

k

)2

= (νt∂U/∂y)2

k2 = νt

k2 Pk = cµε

Pk = cµ (10)

Thus, using the following relation:

Uτ = √τw/ρ √−uv

√c1/2µ k c1/4

µ k1/2P (11)

3 The coefficient cµ can be obtained from the experimental results of local equilibriumboundary layers: −uv/k 0.3, as cµ= (−uv)2/k2 = 0.32 = 0.09.



one can rewrite equation (7) as:

τw = ρc1/4µ k1/2

P UP

1κ

ln (c1/4µ y∗

P) + B(12)

with the relation of y+ = c1/4µ y∗ in local equilibrium [4]. Note that the super-

scripts meaning steps are omitted for simplification hereafter. This form avoidsa shortcoming of using y+ which becomes zero where the wall shear stress τw

vanishes.For turbulence energy k , one can obtain its value at P from equation (11) as:

kP = U 2τ√cµ

(13)

for the boundary condition. The dissipation rate of k is also given as:

εP = U 3τ

κy(14)

for the boundary condition at P by manipulating equations (10) and (11) withthe relation νt/ν= κy+. Note that the friction velocity Uτ(= √

τw/ρ) should beobtained from equation (7) or (12).

9.4 Analytical Wall-Function (AWF)

9.4.1 Basic strategy of the AWF

In the AWF, the wall shear stress and heat flux are obtained through the analyticalsolution of simplified near-wall versions of the transport equations for the wall-parallel momentum and temperature [8]. In case of the forced convection regime, themain assumption required for the analytical integration of the transport equationsis a prescribed variation of the turbulent viscosity µt . The distribution of µt overthe wall-adjacent cell P is modelled as in a one-equation turbulence model:

µt = ρcµk1/2 = ρcµk1/2cy αµy∗ (15)

where is the turbulent length scale, α= ccµ and y∗ ≡ yk1/2P /ν. In order to consider

viscous sub-layer effects, instead of introducing a damping function, the profile ofµt is modelled as:

µt = max0,αµ(y∗ − y∗v ) (16)

in which µt is still linear in y∗ and grows from the edge of the viscous sub-layer:y∗v (≡ yvk

1/2P /ν).



U

yyv

yn

Pn

N

smoothrough

mt = max[0,am(y*−yv*)]

y* ≡ y kp /v

mt

Figure 9.2. Near-wall cells.

In the context of Figure 9.2, the near-wall simplified forms of the momentumand energy equations become

∂

∂y∗

[(µ+ µt)

∂U

∂y∗

]= ν2

kP

[∂

∂x(ρUU ) + ∂P

∂x

]︸︷︷︸

CU

(17)

∂

∂y∗

[(µ

Pr+ µt

Prt

)∂

∂y∗

]= ν2

kP

[∂

∂x(ρU ) + Sθ

]︸︷︷︸

CT

(18)

where Prt is a prescribed turbulent Prandtl number, taken as 0.9.The further assump-tion made is that convective transport and the wall-parallel pressure gradient ∂P/∂xdo not change across the wall-adjacent cell which is a standard treatment in thefinite volume method. Thus, the right-hand side (rhs) terms CU and CT of equa-tions (17) and (18) can be treated as constant. Then, the equations can be integratedanalytically over the wall-adjacent cell giving:

if y∗< y∗v

dU

dy∗ = (CU y∗ + AU )/µ (19)

d

dy∗ = Pr(CT y∗ + AT )/µ (20)

U = CU

2µy∗2 + AU

µy∗ + BU (21)

= PrCT

2µy∗2 + PrAT

µy∗ + BT (22)



if y∗ ≥ y∗v

dU

dy∗ = CU y∗ + A′U

µ1 + α(y∗ − y∗v )

(23)

d

dy∗ = Pr(CT y∗ + A′T )

µ1 + αθ(y∗ − y∗v )

(24)

U = CU

αµy∗ +

A′

U

αµ− CU

α2µ(1 − αy∗

v )

ln [1 + α(y∗ − y∗v )] + B′

U (25)

= PrCT

αθµy∗

(26)

+

PrA′T

αθµ− PrCT

α2θµ

(1 − αθy∗v )

ln [1 + αθ(y∗ − y∗

v )] + B′T

where αθ =αPr/Prt . The integration constants AU , BU , AT , BT etc. are determinedby applying boundary conditions at the wall, yv and the cell face point n. The valuesat n are determined by interpolation between the calculated node values at P andN , whilst at yv a monotonic distribution condition is imposed by ensuring that U , and their gradients should be continuous at y = yv. Notice that to determine theintegration constants the empirical log-law is not referred to at all and the obtainedlogarithmic velocity equation (25) includes CU . The latter implies that the velocityprofile has sensitivity to the pressure gradient since CU includes ∂P/∂x.

The result is that the wall shear stress and wall heat flux can be expressed as:

τw = µ dU

dy

∣∣∣∣w

= µk1/2P

ν

dU

dy∗

∣∣∣∣w

= k1/2P AU

ν(27)

qw = −ρcpν

Pr

d

dy

∣∣∣∣w

= −ρcpν

Pr

k1/2P

ν

d

dy∗

∣∣∣∣w

= −ρcpk1/2P AT

µ(28)

The local generation rate of k , Pk (= νt(dU/dy)2), is written as:

Pk =⎧⎨⎩

0, if y∗ < y∗v

αkP

ν(y∗ − y∗

v )(

CU y∗ + A′U

µ1 + α(y∗ − y∗v )

)2

, if y∗ ≥ y∗v

(29)

which can then be integrated over the wall-adjacent cell to produce an average valuePk for use in solving the k equation of the cell P.

For the dissipation rate ε, the following model is employed:

ε =

2νkP/y2ε , if y < yε

k1.5P /(cy), if y ≥ yε

(30)



Table 9.1. Model coefficients

α c cµ y∗vs y∗

ε Pr∞tccµ 2.55 0.09 10.7 5.1 0.9

U

t

nPUn = (un,vn,wn)

Up = (up,vp,wp)

yn

Pyv

t

n = (nx,ny,nz)

S

Figure 9.3. Skewed near-wall cells.

The characteristic dissipation scale yε can be defined as y∗ε = 2c to ensure a

continuous variation of ε at yε. Thus, the cell averaged value is obtained as:

ε =

⎧⎪⎨⎪⎩2k2

P/(νy∗2ε ), if y∗

ε > y∗n

1

yn

(yε

2νkP

y2ε

+∫ yn

yε

k1.5P

cydy

)= k2

P

νy∗n

[2

y∗ε

+ 1

cln

(y∗

n

y∗ε

)], if y∗

ε ≤ y∗n

(31)

Through numerical experiments, the value of the constant y∗v was optimised to be

10.7 which corresponds to approximately half the thickness of the conventionallydefined viscous sub-layer of y+ = 11. The other model coefficients are listed inTable 9.1.

9.4.2 AWF in non-orthogonal grid systems

In application codes which are written in either structured or unstructured gridmethod, the near-wall cells are not always orthogonal to the walls as shown inFigure 9.3. Thus, the way of obtaining the cell face values, Un and yn, for the AWFapproach is as follows.

When the velocity vector−→UP at the node P is decomposed to the normal and

tangential directions of the wall, they can be written as:−→U n

P = (−→UP · −→n )−→n (32)

−→U t

P = −→UP − −→

U nP (33)

Since the tangential unit vector is obtained as−→t = −→

U tP/|

−→U t

P|, the velocity at theface n can be obtained as:

Un = −→t · −→

Un (34)



Although the distance yn of a tetrahedral or pyramidal (triangle in 2D) cell isdetermined as a wall-normal distance from its apex, that of a hexahedral or prismatic(quadrilateral in 2D) cell is obtained by

yn = Vol/S (35)

where Vol and S are the volume and the wall area of the cell, respectively.When one calculates the velocity components, the wall shear stress τw

−→t should

be decomposed to each direction.

Examples of applicationsAlthough theAWF has shown its superb performance in many flow fields [8, 17–19],the results of a relatively complicated case are shown here.

Figure 9.4a shows the geometry and the mesh used for an IC Engine port-valve-cylinder flow [18, 19]. The total cell number of 300,000 is applied for a half model.The flow comes through the intake port and the valve section into the cylinder atRe 105. The turbulence model used is the standard k-ε model.

As shown in Figure 9.4b, it is clear that the standard LWF produces too largek distribution in the region between the valve and the valve-seat circled by brokenlines.TheAWF, however, reduces such production due to its inclusion of the pressuregradient dependency. This leads to the improved distribution of the gas dischargecoefficient Cd by the AWF as shown in Figure 9.5 though it is not very significant.The tumble ratio, which is the momentum ratio of the in-cylinder longitudinal vortexand the inlet flow, at the point of 6.8 mm of the valve lift is also better predicted bythe AWF.

9.4.3 AWF for rough wall turbulent flow and heat transfer

In a rough wall turbulent boundary layer, as shown in Figure 9.6, the logarithmicvelocity profile shifts downward depending on the equivalent sand grain roughnessheight h and written as an empirical formula [20]:

U+ = 1

κln

y+

h+ + 8.5 (36)

for completely rough flows. This implies that the flow becomes more turbulent dueto the roughness. It is thus considered that the viscous sub-layer is destroyed in the‘fully rough’ regime while it partly exists in the ‘transitional roughness’ regime.

In common with conventional wall functions (e.g. Cebeci and Bradshaw [21]),this extension of theAWF strategy [10] to flows over rough surfaces involves the useof the dimensionless roughness height. In this case, however, instead of h+, h∗ isused to modify the near-wall variation of the turbulent viscosity. More specifically,in a rough wall turbulence, y∗

v is no longer fixed at 10.7 and is modelled to becomesmaller. This provides that the modelled distribution of µt shown in Figure 9.2shifts towards the wall depending on h∗. At a certain value of the dimensionless



Standard k-ε + LWF Standard k-ε + AWF

(b)

(a)

1009080706050403020100

Figure 9.4. Port-valve-cylinder flow: (a) mesh, (b) k distribution.

roughness height h∗ = A, y∗v = 0 is assumed and at h∗>A it is expediently allowed

to have a negative value of y∗v to give a positive value of µt at the wall as:

y∗v = y∗

vs1 − (h∗/A)m (37)

where y∗vs is the viscous sub-layer thickness in the smooth wall case. The optimised

values for A and m have been determined through a series of numerical experimentsand comparisons with available flow data. The resultant form is:

y∗v = y∗

vs1 − (h∗/70)m (38)



0.8

LDA

LWF

Exp

Cd

Tumble

0.53

1.28

0.577

1.046 0.96

0.589

AWF LWF

AWF

0.7

0.6

0.5

Cd

Dis

char

ge c

oeffi

cien

t

0.4

0.3

0.2

0.1

02.10 3.16 4.21 5.26

Valve lift [mm]

6.31 7.37 8.42 9.47

Figure 9.5. Distribution of gas discharge coefficient.

Smooth

Rough

log (y+)

U +

Figure 9.6. Surface roughness effects on velocity profile.

with

m = max

(0.5 − 0.4

(h∗

70

)0.7)

,

(1 − 0.79

(h∗

70

)−0.28)

(39)

For h∗< 70, the viscosity-dominated sub-layer exists, but with the above modi-fied value for y∗

v . When h∗> 70, corresponding to y∗v < 0, it is totally destroyed.

Note that although the experimentally known condition for the full rough regime ish+ ≥ 70, the condition when the viscous sub-layer vanishes from equation (38) ish∗ = 70 that corresponds to h+ 40 in a typical channel flow case.This is rather con-sistent with the relation of the viscous sub-layer thicknesses in the smooth wall case.(In the smooth wall case, the viscous sub-layer thickness of y∗

v = 10.7 correspondsto approximately half the thickness of the conventionally defined viscous sub-layerof y+ = 11.) It is thus considered that a part of the transitional roughness regime issomehow effectively modelled into the region without the viscous sub-layer by thepresent strategy due to simply assuming the near-wall turbulence behaviour.



Unlike in a sub-layer over a smooth wall, the total shear stress now includes thedrag force from the roughness elements in the inner layer which is proportional tothe local velocity squared and becomes dominant away from the wall, compared tothe viscous force. In fact, the data reported by Krogstad et al. [22] and Tachie et al.[23] showed that the turbulent shear stress away from the wall sometimes becamelarger than the wall shear stress. This implies that the convective and pressuregradient contributions should be represented somewhat differently across the innerlayer, below the roughness element height. Hence, the practice of simply evaluatingthe rhs of equation (17) in terms of nodal values needs modifying. In the presentstrategy a simple approach has been taken by assuming that the total shear stressremains constant across the roughness element height. Consequently, one is led to

CU =⎧⎨⎩

0, if y∗ ≤ h∗ν2

kP

[∂

∂x(ρUU ) + ∂P

∂x

], if y∗ > h∗ (40)

In the energy equation, Prt is also no longer constant over the wall-adjacent cell.The reason for this is that since the fluid trapped around the roots of the roughnesselements forms a thermal barrier, the turbulent transport of the thermal energy iseffectively reduced relative to the momentum transport. (The results of the rib-roughened channel flow direct numerical simulation (DNS) by Nagano et al. [24]support this consideration since their obtained turbulent Prandtl number increasessignificantly towards the wall in the region between the riblets.) Thus, as illustratedin Figure 9.7, although it might be better to model the distribution of Prt with a non-linear function, a simple linear profile is assumed in the roughness region of y ≤ h as:

Prt = Pr∞t +Prt (41)

Prt = Ch max (0, 1 − y∗/h∗) (42)

After a series of numerical experiments referring to the experimental correlation,the following form for Ch has been adopted within the roughness elements (y ≤ h):

Ch = 5.51 + (h∗/70)6.5 + 0.6 (43)

Over the rest of the field (y> h), Prt = Pr∞t is applied. (See Table 9.1 for the modelcoefficients.) Note that since the turbulent viscosity is defined as zero in the regiony< yv, the precise profile adopted for the turbulent Prandtl number in the viscoussub-layer (y< yv) does not affect the computation.

At a high Pr flow (Pr>> 1), the sub-layer, across which turbulent transport ofthermal energy is negligible, becomes thinner than the viscous sub-layer. Thus,the assumption that the turbulent heat flux becomes negligible when y< yv nolonger applies. (See the high Pr version of the AWF for such flows presented insection 9.4.5.)

The analytical solutions of both mean flow and energy equations then can beobtained in the four different cases illustrated in Figure 9.8 assuming that the wall-adjacent cell height is always greater than the roughness height. Although one can



Prt

y*v h* y*

Prt∞

∆Prt

Figure 9.7. Modelled turbulent Prandtl number distribution.

N

(a)

n

P

h

N

(b)

n

P

hyv

yv

yv

N

(c)

n

P

h

N

(d)

n

P

h

Figure 9.8. Near-wall cells over a rough wall: (a) yv≤ 0, (b) 0< yv≤ h,(c) h< yv≤ yn, (d) yn< yv.

apply the models of equations (40) and (41) at any node point, limiting them tothe wall-adjacent cells is preferable from the numerical view point since it onlyrequires a list of the cells facing to walls for wall boundary conditions. (Despitethat, in simple flow cases, the AWF is still applicable to wall-adjacent cells whoseheight is lower than the roughness height [10].) Even in the case without any viscoussub-layer, case(a) of Figure 9.8a, the resultant expressions for τw and qw are of thesame form as those of equations (27) and (28). However, different values of AU

and AT , which are functions of the roughness height, are obtained, correspondingto the four different cases. (See Appendix A for the detailed derivation process.)

In Table 9.A1, the cell averaged generation term Pk and AU are listed, intro-ducing Y ∗ ≡ 1 +α(y∗ − y∗

v ). Note that equations (30) and (31) are still used for thedissipation, and the integration for Pk in Table 9.A1 can be performed as follows:

∫ b

a(y − yv)

(Cy + A

1 + α(y − yv)

)2

dy

=[

C2

2α2 y2 + C(2A + Cyv − 2C/α)α2 y + (A + Cyv − C/α)2

α2[1 + α(y − yv)](44)

+ (A + Cyv − C/α)(A + Cyv − 3C/α)α2 ln [1 + α(y − yv)]

]b

a



0.01

0.1

1,000 10,000 100,000 1e + 06 1e + 07 1e + 08

f

Re

Moody (1944)

AWF cal.

h/D5 × 10−2

3 × 10−2

2 × 10−2

1 × 10−2

5 × 10−3

2 × 10−3

1 × 10−3

5 × 10−4

1 × 10−4

5 × 10−5

1 × 10−5

0

Figure 9.9. Friction factors in pipe flows (Moody chart).

For heat transfer, the resultant form of the integration constant AT can bewritten as:

AT = µ( n − w)/Pr + CT ET /DT (45)

where the coefficients DT and ET are listed in Table 9.A2, defining αT ≡αPr/(Pr∞t ), βT ≡ C0/(h∗Pr∞t ), Y αT ≡ 1 +αT (y∗ − y∗

v ), Y βT ≡ 1 +βT (y∗ − y∗v ),

and λb ≡ Y αT0 +βT h∗.

In the case of a constant wall heat flux condition, the wall temperature is obtainedby rewriting equations (28) and (45) as:

w = n + Prqw

ρcpk1/2P

DT + PrCT ET

µ(46)

Examples of applicationsThe AWF has been implemented with the ‘standard’ linear k-ε (Launder andSpalding [4], Launder and Sharma: LS [25]) and also with the cubic non-lineark-ε model of Craft, Launder and Suga: CLS [26]. (Although the LS and the CLSmodels are LRN models, with the wall-function grids, the LRN parts of the modelterms do not contribute to the computation results.) For the turbulent heat flux inthe core region, the usual eddy diffusivity model with a prescribed turbulent Prandtlnumber Prt = 0.9 is used.

Pipe flows. Figure 9.9 compares the predicted friction coefficient and the exper-imental correlation for turbulent pipe flows, known as the Moody chart [27]. Theturbulence is modelled by the linear k-ε model with the AWF. In the range ofh/D = 0 to 0.05 (D: pipe diameter) and Re = 8000 to 108, it is confirmed thatthe AWF performs reasonably well over a wide range of Reynolds numbers androughness heights.



300 398

40019°

R 1

200160

Airy

x

Figure 9.10. Flow geometry of the convex rough wall boundary layers of Turneret al. [28].

Curved wall boundary layer flows. Heat transfer along curved surfaces is verycommon and important in engineering applications such as in heat sinks and aroundturbine blades. Thus, although the AWF itself does not explicitly include sensitivityto streamline curvature, it is useful to confirm its performance when applied in com-bination with a turbulence model which does capture streamline curvature effects.Hence, the turbulence model used here is the cubic non-linear k-ε model (CLS).

For comparison, the rough wall heat transfer experiments over a convex surfaceby Turner et al. [28] are chosen. The flow geometry is shown in Figure 9.10. Theworking fluid was air at room temperature and the wall was isothermally heated.The comparison is made in the cases of trapezoidal-shaped roughness elements.According to Turner et al., the equivalent sand grain roughness height h is 1.1 timesthe element height. The computational mesh used is 90 × 20 whose wall adjacentcell height is 5 mm, which is greater than the equivalent sand grain roughnessheights. (A fine mesh of 90 × 50 whose wall adjacent cell height is 1mm is alsoused to confirm the sensitivity to the wall-adjacent cell heights. The correspondingdiscussion is addressed in the following paragraph.)

Figure 9.11 compares the heat transfer coefficient α distribution under zero-pressure gradient conditions. Figure 9.11a shows the case of h = 0.55 mm. The inletvelocities of U0 = 40, 22 m/s, respectively, correspond to h+ 90, 50 and thus theyare in the full and the transitional roughness regimes. In the case of h = 0.825 mm,shown in Figure 9.11b, U0 = 40, 22 m/s correspond to h+ 135, 80 which are bothin the fully rough regime. From the comparisons, although there can be seen somediscrepancy, it is recognised that the agreement between the experiments and thepredictions is acceptable in both the full and transitional roughness regimes. InFigure 9.11b, the result by the fine mesh of 90 × 50 whose wall adjacent cell heightis 1 mm is also plotted for the case of U0 = 22 m/s. It is readily seen that the AWFis rather insensitive to the near-wall mesh resolution in such a flow case since twoprofiles from very different resolutions are nearly the same.

Figure 9.11 also shows the effects of the wall curvature on the heat transfercoefficients in the case of U0 = 40 m/s. (The curved section is in the region of300 mm ≤ x ≤ 698 mm.) Although the curvature effects observed are not large,since the curvature is not very strong, they are certainly predicted by the computa-tions with the AWF and cubic non-linear k-ε model. Turner et al. reported that thecurvature appeared to cause an increase of 2 to 3% in the heat transfer coefficient α.



(a)

0

50

a [W

/(m2 K

)]a

[W/(m

2 K)]

100

150

200

250

300

350

400

0 100 200 300 400 500 600 700x(mm)

Element height = 0.5 mm: h = 0.55 mmExpt. AWF + CLS

:U0 = 40 m/s :U0 = 22 m/s :U0 = 40 m/s; straight

(b)

0

50

100

150

200

250

300

350

400

0 100 200 300 400 500 600 700

x(mm)

Element height = 0.75 mm: h = 0.825 mmExpt. AWF + CLS

:U0 = 40 m/s :U0 = 22 m/s :U0 = 40 m/s; straight:U0 = 22 m/s; fine mesh

Figure 9.11. Heat transfer coefficient distribution in convex rough wall boundarylayers at Pr = 0.71.

Computations are consistent with this experimental observation. Note that thepresent curvature effects are of the entrance region of the convex wall. Due tothe sudden change of the curvature rate, the pressure gradient becomes locallystronger near the starting point of the convex wall resulting in flow acceleration andthus heat transfer enhancement.

Sand dune flow. Figure 9.12 shows the geometry and a computational mesh usedfor a water flow over a sand dune of van Mierlo and de Ruiter [29]. Their exper-imental rig consisted of a row of 33 identical 2D sand dunes covered with sandpaper, whose averaged sand grain height was 2.5 mm. Since they measured theflow field around one of the central dunes of the row, streamwise periodic boundaryconditions are imposed in the computation. In the experiments, the free surface



−0.1−0.05

00.050.1

0.150.2

0.250.3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

y (m

)

x (m)

165 × 25

Water flow

160 100 250 7501600

260 80

8

y

x

292

80 80 72

Figure 9.12. Dune profile and computational grid.

−0.05

0.0

0.05

0.1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Uτ/

Ub

x (m)

Expt.

Grid 165 × 25 (LWF + CLS)

Grid 329 × 50 (AWF + CLS)



Figure 9.13. Friction velocity over the sand dune.

was located at y = 292 mm and the bulk mean velocity was Ub = 0.633 m/s. Thus,the Reynolds number based on Ub and the surface height was 175,000. In this flow,a large recirculation zone appears behind the dune. Due to the flow geometry, theseparation point is fixed at x = 0 m as in a back-step flow.

Figure 9.13 compares the predicted friction velocity distribution on meshes of330 × 50, 165 × 50 and 165 × 25. In the finest and coarsest meshes, the heights ofthe first cell facing the wall are, respectively, 4 and 8 mm, and larger than the grainheight of 2.5 mm which corresponds to h+ 75 at the inlet. The turbulence model



applied is the cubic CLS k-ε model with the differential length-scale correctionterm for the dissipation rate equation [30]. The lines of the predicted profiles arealmost identical to one another, proving that the AWF is rather insensitive to thecomputational mesh. (In the following discussions on flow field quantities, resultsby the finest mesh of 330 × 50 are used.)

Figure 9.13 also shows the result by the LWF. Although the LWF, equation (36)should perform reasonably in flat plate boundary layer type of flows, it is obviousthat the LWF produces an unstable wiggled profile in the recirculating region of0 m< x< 0.6 m. The computational costs by the AWF and the LWF are almostthe same as each other, with the more complex algebraic expressions of the AWFrequiring slightly more processing time.

Figure 9.14 compares flow field quantities predicted by the CLS and the LSmodels with the AWF. In the distribution of the mean velocity and the Reynoldsshear stress, both models agree reasonably well with the experiments as shown inFigure 9.14a and b while the CLS model predicts the streamwise normal stressbetter than the LS model (Figure 9.14c). These predictive trends of the models areconsistent with those in separating flows by the original LRN versions and thus itis confirmed that coupling with the AWF preserves the original capabilities of theLRN models.

Ramp flow. Figure 9.15 illustrates the flow geometry and a typical computationalmesh (220 × 40) used for the computations of the channel flows with a ramp onthe bottom wall by Song and Eaton [31]. A 2D wind tunnel whose height was152 mm with a ramp (height H = 21 mm, length L = 70 mm, radius R = 127 mm)was used in their experiments. Air flowed from the left at a free stream velocityUe = 20 m/s with developed turbulence (Reθ = 3,400 at x = 0 mm for the smoothwall case; Reθ = 3,900 for the rough wall case). Since for the rough wall case sandpaper with an averaged grain height of 1.2 mm, which corresponds to h+ 100 atx = 0 mm, covered the ramp part, the height of the first computational cell from thewall is set as 1.5 mm. (Note that since the location of the separation point is notfixed in this flow case, the grid sensitivity test, which is not shown here, suggeststhat unlike in the other flow cases large wall adjacent cell heights affect predictionof the recirculation zone.) This flow field includes an adverse pressure gradientalong the wall and a recirculating flow whose separation point is not fixed, unlikein the sand dune flow. The measured velocity fields implied that the recirculationregion extended between 0.74 ≤ x′(: x/L) ≤ 1.36 and 0.74< x′ ≤ 1.76 in the smoothand rough wall cases, respectively. Thus, relatively finer streamwise resolution isapplied to the computational mesh around the ramp part.

Figure 9.16 compares the predicted pressure coefficients of the smooth wallcase by the AWF and the standard LWF with the results of the LRN computation.The turbulence model used is the CLS model. The first grid nodes from the wall(y1) are located at y+

1 = 15 ∼ 150 for the wall-function models while those for theLRN are located at y+

1 ≤ 0.2. (The mesh used for the LRN case has 100 cells inthe y direction.) Obviously, theAWF produces profiles that are closer to those of theexperiment than the profiles of the LWF model. The AWF reasonably captures the



(c)

−0.1−0.05

00.050.1

0.150.2

0.250.3

x (m)

0 0.2 0.4 0.6 0.8 1 1.2

0.0 0.3

1.4 1.6 1.8

Expt.AWF+ CLSAWF + LS

1.58x = 0.06 0.21 0.37 0.60 0.82 0.97 1.12 1.27 1.42

uu /Ub

(b)

−0.1−0.05

00.050.1

0.150.2

0.250.3

x (m)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

1.58x = 0.06 0.21 0.37 0.60 0.82 0.97 1.12 1.27 1.42

0.0 0.03

Expt.AWF + CLSAWF + LS

−uν/U2

(a)

−0.1−0.05

00.050.1

0.150.2

0.250.3

x (m)

y (m

)y

(m)

y (m

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

1.58x = 0.06 0.21 0.37 0.60 0.82 0.97 1.12 1.27 1.42

0.0 1.5

Expt.AWF + CLSAWF + LS

U/Ub

b

Figure 9.14. Mean velocity and Reynolds stress distribution of the sand dune flow.

effects of adverse pressure gradients due to the inclusion of the sensitivity to pressuregradients in its form. Hence, its predictive tendency is similar to that of the LRNmodel.

Figure 9.17a compares the mean velocity profiles for both the smooth andrough wall cases. For both the cases, the agreement between the experimentsand the predictions by the AWF coupled to the CLS model is reasonably good.



L (70)

R (1

27)

H (

21)

152131

Flow

x

y

020406080

100120140160

−200 −100 0 100 200 300 400 500

y (m

m)

x (mm)

220 × 40

Figure 9.15. Ramp geometry and computational grid.

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

−2 −1 0 1 2 3 4 5 6 7

Cp

x ′

Top wall (smooth), Expt.

Bottom wall (smooth), Expt.

AWF y1+= 15∼150

LWF y1+= 15∼150

LRN y1+= ∼0.2

Figure 9.16. Pressure coefficient distribution of the smooth ramp flow; CLS is usedin all the computations.

The predicted recirculation zones are 0.69< x′< 1.40 and 0.57< x′< 1.60 in thesmooth and rough wall cases, respectively. They, however, do not correspond wellwith the experimentally estimated ones. Figure 9.17b–d compare the distributionof the Reynolds stresses at the corresponding locations to those in Figure 9.17a.(The values are normalised by the reference friction velocity Uτ,ref at x′ = −2.0.)The agreement between the prediction and the experiments is again reasonably goodfor each quantity.



x’= –2.00 0.00 0.50 0.74 1.00 1.36 1.76 2.00 4.00 7.00

(b)

–1

–0.5

0

0.5

1

1.5

2

2.5

y/H

–uν/U2τ,ref

Expt.(rough)Expt.(smooth)AWF+CLS(rough)AWF+CLS(smooth)

0.0 3.0

x’= –2.00 0.00 0.50 0.74 1.00 1.36 1.76 2.00 4.00 7.00

(c)

–1

–0.5

0

0.5

1

1.5

2

2.5

y/H

√uu/Uτ,ref


0.0 3.0

x’= –2.00 0.00 0.50 0.74 1.00 1.36 1.76 2.00 4.00 7.00

(d)

–1

–0.5

0

0.5

1

1.5

2

2.5

y/H

√νν/Uτ,ref


0.0 3.0

(a)

–1

–0.5

0

0.5

1

1.5

2

2.5y/

H

x’= –2.00 0.00 0.50 0.74 1.00 1.36 1.76 2.00 4.00 7.00


0.0 1.5U/Ue

Figure 9.17. Mean velocity and Reynolds stress distribution of the ramp flow.



9.4.4 AWF for permeable walls

Flows over permeable walls can be seen in many industrially important devicessuch as catalytic converters, metal foam heat exchangers and separators of fuelcells. Some geophysical fluid flows, i.e. water flows over river beds etc., are oftenconcerned as flows over permeable surfaces as well.

The effects of a porous medium on the flow in the interface region between theporous wall and clear fluid (named ‘interface region’ hereafter) have been focusedon by many researchers. Due to the severer complexity of the flow phenomena,however, experimental turbulent flow studies in the interface regions are rather lim-ited. Zippe and Graf [32] and others experimentally found that the friction factorsof turbulent flows over permeable beds became higher than those over imperme-able walls with the same surface roughness. The recent DNS study of the interfaceregion by Breugem et al. [33] also indicated that the friction velocity became higherassociated with the increase of the permeability. All these results imply that thewall permeability effects on turbulence should be solely taken into account for flowsimulations.

This part, thus, introduces an attempt [11] to construct an AWF which includesthe effects of the wall permeability as well as the wall roughness for computing theinterface regions over porous media.

Modelling for porous/fluid interfacial turbulenceIt is natural to consider a slip tangential velocity Uw at the interface between aclear fluid flow and a porous surface when one introduces a volume averagingconcept. Before discussing the slip velocity, it is essential to define a nominallocation of the interface. In some porous media, their outermost perimeter maybe reasonable as the interface [34]. This means when the pores are filled withsolid material, a smooth impermeable boundary is recovered. However, in case thata porous medium is composed of beads, the recovered impermeable surface hassome roughness by defining the pores as the spaces surrounded by the beads. Thisdefinition is consistent with those in the experiments by Zippe and Graf [32] and

U

Uw

y

x

P

N

n

h y = 0

Hh

(b)(a)

UdPorousmedium

Figure 9.18. (a) Velocity profile of a channel flow over a porous medium, (b) near-wall cell.



others. Therefore, the surface roughness h of porous/fluid interface is consideredas in Figure 9.18a and that the interface is located at the bottom of the roughnesselements. In order to obtain the tangential velocity Uw at the interface (y = 0), thevelocity distribution in a porous medium is estimated as follows.

Following Whitaker [35], the volume averaged Navier-Stokes equations forincompressible flows in isotropic porous media using the decomposition:

ui = 〈ui〉 f + ui (47)

are:

∂〈ui〉 f

∂t+ 〈uj〉 f ∂〈ui〉 f

∂xj= − 1

ρ

∂〈p〉 f

∂xi+ ν∂

2〈ui〉f

∂x2j

− ∂τij

∂xj+ fi (48)

∂〈ui〉∂xi

= 0 (49)

where p, ρ, ν, and ϕ are, respectively, the pressure, the density, the kinematicviscosity of the fluid and the porosity of the porous medium. The volume averagedvalues 〈φ〉 and 〈φ〉f are superficial and intrinsic averaged values of a variable φ,respectively. (The superficial averaging is defined by taking average of a variableover a volume element of the medium consisting of both solid and fluid materials,while the intrinsic averaging is defined over a volume consisting of fluid only.)Between them the following relation exists:

〈φ〉 = ϕ〈φ〉 f (50)

Although the sub-filter-scale dispersion ∂τij/∂xj needs a closure model as in largeeddy simulations, it could be neglected in porous media since it is small enoughcompared with the other terms. As in Whitaker [35], the drag force fi in isotropicporous media can be parametrised as:

fi −ν 〈uj〉Kij

− νFij〈uk〉Kjk

(51)

where Kij and Fij are, respectively, the permeability tensor and the Forchheimercorrection tensor. These tensors are empirically given by Macdonald et al. [37]being based on the modified Ergun equation [36] as:

Kij = d2pϕ

3

180(1 − ϕ)2︸︷︷︸K

δij (52)

Fij = ϕ

100(1 − ϕ)

dp

ν︸︷︷︸F

√(〈uk〉 f )2δij (53)

where dp is the mean particle diameter.



Thus, by the Reynolds averaging of the volume averaged momentum equa-tions one can obtain the following equations for a developed turbulent flow in ahomogeneous porous medium:

0 = − 1

ρ

∂P

∂x− ∂uv

∂y+ ν

ϕ

∂2U

∂y2 − ν

KU − νϕF

Kq〈u〉 f (54)

0 = − 1

ρ

∂P

∂y− ∂v2

∂y− νϕF

Kq〈v〉 f (55)

where q =√〈uk〉 f 〈uk〉 f , uiuj = (〈ui〉 f − 〈ui〉 f )(〈uj〉 f − 〈uj〉 f ), U = 〈u〉, and

P =〈p〉f .In laminar flow cases where the permeability Reynolds number ReK ≡√

KUτ/ν, which is based on the permeability K and the friction velocity Uτ , issufficiently small, the Reynolds stress and the Forchheimer drag terms can beneglected:

0 = − 1

ρ

∂P

∂x+ ν

ϕ

∂2U

∂y2 − ν

KU (56)

0 = − 1ρ

∂P

∂y(57)

The solution of the above equations (56) and (57), namely, the Brinkman equations[38] is a decaying exponential function which meets the interfacial (slip) velocityUw at y = 0 and the Darcy velocity Ud deep inside the porous medium.

U = Ud + (Uw − Ud ) exp(

y

√ϕ

K

)(58)

The Darcy velocity is:

Ud = −K

µ

∂P

∂x(59)

In the case of high ReK , turbulent diffusion and Forchheimer drag terms in themomentum equations should be considered and thus their general analytical solutioncannot be easily obtained. Breugem et al. [33] found that a good approximationof the turbulent flow inside the porous media can be obtained by the followingexponential formula:

U Ud + (Uw − Ud) exp(αpy

√ϕ

K

)(60)

where αp is an empirical coefficient. If ReK is small enough, αp = 1. In very largeReK cases, where the turbulent diffusion and the Forchheimer drag terms should



balance each other out:

−∂uv∂y

νϕF

Kq〈u〉f (61)

The Reynolds stress estimated with the eddy viscosity (νt uty; ut is an appropriatevelocity scale) and the velocity gradient from equation (60) leads to

−uv = νt1

ϕ

∂U

∂y uty

U

ϕαp

√ϕ

K αpu2

t y

√ϕ

K(62)

then the LHS of equation (61) may be rewritten as

−∂uv∂y

αp

√ϕ

Ku2

t (63)

Thus, αp can be estimated as

αp

√ϕ

Ku2

t νϕF

Ku2

t ⇒ αp ν√ϕ

KF (64)

Therefore, the following form can be a model for αp:

αp = fαp + (1 − fαp )ν

√ϕ

KF (65)

fαp = exp(

−ReK

βp

)(66)

bridging between 1 and ν√ϕ/KF depending on the permeability. By referring to the

distribution of the coefficient αp obtained from the DNS [33], the model coefficientβp can be functionalised as

βp = 90

Re0.68K

(67)

Hence

fαp = exp

(−Re1.68

K

90

)(68)

(Note that the above form is rewritten in a different way for the AWF as in equation(73).)

Consequently, the slip velocity Uw can be expressed as

Uw = Ud + ∂U/∂y|y=0

αp√ϕ/K

(69)

with the velocity gradient ∂U/∂y|y=0 given by the AWF.



In the AWF, the wall shear stress is obtained by the analytical solution of a sim-plified near-wall version of the transport equation for the wall-parallel momentum.The resultant shear stress form here is the same as that for the rough wall turbulencebut with the slip velocity Uw in the integration constant AU .

Since the wall permeability also makes the flow more turbulent, y∗v should have

its dependency. Equation (38) is thus further modified as

y∗v = y∗

vs

1 − (h∗/70)m − fK

(70)

where fK is a function of the permeability. By referring to the experiments andthe DNS data [32, 33], the range of y∗

v which depends on the wall permeabilityis estimated. Considering the roughness effects in y∗

v , the optimised function inequation (70) for the permeability effects is

fK = 3.7

[1 − exp

−

(K∗

6

)2]

(71)

where K∗ = √KkP/ν. In this form, K∗ is used rather than ReK since the AWF uses√

kP for the velocity scale. In order to keep consistency with such normalisation,Equation (68) is rewritten as

fαp exp(

−K∗1.68

250

)(72)

using the relation of ReK c1/4µ K∗. The following modified form, however, is

employed after further tuning of the model function

fαp = exp(

−K∗1.76

180

)(73)

Examples of applicationsPermeable-wall channel flows. In order to confirm the performance of the method,computations of fully developed turbulent flows in a plane channel with a permeablebottom wall at y/H = 0 and a solid top wall at y/H = 1 (see Figure 9.18a) are shownhere. The considered flow conditions are the same as those in the DNS [33]. Themean diameter of the particle composing the permeable beds is dp/H = 0.01 andtheir porosity are ϕ= 0.95, 0.80, 0.60, 0.00. The permeability and the Forchheimerterm are then obtained by equations (52) and (53).The bulk flow Reynolds number isReb = 5, 500 in all the cases. The roughness height is considered to be h = dp in thispractice. The grid cell number used is only 10 in the wall normal direction due to theuse of the wall-function approach for such relatively low Reynolds number flows.The turbulence models used are the standard k-εmodel [4] and the two-componentlimit (TCL) second moment closure of Craft and Launder [39].

Figure 9.19a compares the mean velocity distribution inside the porous walls.The obtained velocity profiles are from equation (60). Whilst there can be seen



0.3 1.5

1.0

0.5

0.95

w = 0.80

w = 0.95

0.00.80 Re = 5500

0.0 0.2 0.4 0.6 0.8 1.0

Present

Present

DNS

DNS

0.2

0.1

0.0

y/H y/H

(b)(a)

−0.15 −0.10 −0.05 0.00

U/U

b

U/U

bFigure 9.19. Mean velocity distribution: (a) inside porous walls, (b) channel region.

a margin to be improved in the comparison between the present and the DNSresults in the case of ϕ= 0.95, the agreement is virtually perfect in the case ofϕ= 0.80.

The mean velocity distribution in the channel region is compared in Figure9.19b. It is obvious that the computation well reproduces the characteristic velocityprofiles near the permeable walls.

The AWF is also applicable to the second moment closures. Figure 9.20a–ccompares the obtained Reynolds stress distribution by the second moment closure[39]. Due to the use of the AWF, it seems hard to capture the near wall peaks of theReynolds normal stresses as in Figure 9.20a and b. However, general tendencies ofthe distribution profiles are satisfactorily reproduced by the present scheme.

Turbulent permeable boundary layer flows. The other test cases are turbulentboundary layer flows over permeable walls. The same flow conditions of the exper-iments of Zippe and Graf [32] are imposed in the computations. The free stream

turbulence is√

u2/U0 = 0.35×10−2. The Reynolds number based on the momen-tum thickness ranges Reθ = 1.3 − 2.1 × 104. Although it was not reported in Ref.[32], the permeability can be estimated by equation (52) as K = 2.54 × 10−9 m2

since the experimentally used mean diameter of the ellipsoidal beads composingthe permeable bed was dp = 2.883 mm. The presently applied porosity is ϕ 0.30considering the effects of non-spherical shapes of beads used in the experiments.(The porosity of a fully packed body-centred cubic structure by spherical beadsis ϕ = 0.26. Zippe and Graf did not report about the porosity as well.) The gridcell number used is 120 × 150 for the developing boundary layer computations.The turbulence model used is the standard k-εmodel of Launder and Spalding [4].Since Zippe and Graf [32] also did not report the roughness height of their testcases, it is presently estimated as h+ 116 comparing the experimental velocityprofile with the formula of Nikuradse [20]: Equation (36).

Figure 9.21 compares the mean velocity profiles where their boundary layerthicknesses correspond to each other. The cases are R (non-permeable rough



3w = 0.95

w = 0.95

w = 0.80

0.80

0.0

DNS : DNS

present :

present

2

1

0

−10.0

(a) (b)

(c)

0.2 0.4 0.6 0.8 1.0

4

3

2

1

0

0

0.0 0.2 0.4 0.6 0.8 1.0y/H y/H

4

3

2

1

0.0 0.2 0.4 0.6 0.8 1.0y/H

uv U2 τ

u2/Uτ√v2/Uτ√

w2/Uτ√

DNS presentu2/Uτ√v2/Uτ√

w2/Uτ√

Figure 9.20. Reynolds stress distribution in permeable channel flows.

Expt. present

20

10

0100 1,000

y +

U +

10,000

RP12.5

P16.5

Figure 9.21. Mean velocity distribution in permeable boundary layers.

wall boundary layer), P12.5 (permeable wall boundary layer with U0 = 12.5 m/s)and P16.5 (permeable wall boundary layer with U0 = 16.5 m/s). Obviously, thepresent scheme reasonably reproduces the experimentally obtained velocity profileswith/without the permeability.



0

5

10

15

20

25

1 10 100 1,000 10,000

y+

Re = 105 Pr = 0.71

0.2

0.1

0.025

LS(Pr = 0.71)k-ε + AWFKader

Θ+

Figure 9.22. Mean temperature profiles in turbulent smooth channel flows at Pr< 1.

9.4.5 AWF for high Prandtl number flows

If one considers to predict turbulent wall heat transfer of high Prandtl number fluidflows such as cooling oil and IC engine water-jacket flows, it is essential to analysethe thermal boundary layer which is much thinner than that of the flow boundarylayer. Thus, near-wall modelling which resolves the viscous sub-layer has beenthought to be essential for high Pr thermal fields. For example, at the developmentof a novel turbulent heat flux model applicable to general Pr cases, Rogers et al. [40]supposed correct near-wall stress distribution and Suga and Abe [41] employed anLRN nonlinear k-ε model. So and Sommer [42] also applied an LRN k-ε as wellas a near-wall stress transport model for flows at Pr = 1000.

However, even with the recent development of LRN heat transfer models,wall-function approaches still attract industrial engineers. Its reasons are a highcomputational cost of the LRN computation and difficulty of generating qualitynear-wall grids for complex 3D flow fields such as an IC engine water jacket.

Although theAWF performs reasonably well at Pr< 1 as shown in Figure 9.22,4

its applicability to higher Pr cases has not been discussed well so far. Therefore, thispart focuses on the improvement of the thermal AWF for high Pr turbulent flowswith and without wall roughness [12].

High Pr AWF for smooth wall heat transferIn the AWF for smooth wall heat transfer, µt variation is assumed thatµt is zero fory∗ ≤ y∗

v = 10.7 and then increases linearly as of equation (16). Since the theoretical

4 In the cases shown in Figure 9.22, the constant Prt = 0.9 is used for convenience. However,for lower Pr cases, it is well known that higher values of Prt should be used [43]. There isthus a tendency for the AWF to underpredict the mean temperature profile, particularly, atPr = 0.025.



a′my∗3/Prt

m/Pr :(air)

m/Pr : (oil)

mt /Prt

y∗ν y∗

b

The

rmal

diff

usiv

ities

am(y∗−y∗ν)/Prt

Figure 9.23. Near-wall thermal diffusivity distribution.

wall-limiting variation of µt is proportional to y3, this assumption does not count acertain amount of turbulent viscosity in the viscous sub-layer. Despite that, its effectis not serious for flow field prediction since the contribution from the molecularviscosity is more significant in the sub-layer. This is also true for the thermal fieldprediction of fluids whose Pr is less than 1.0. However, in high Pr fluid flows such asoil flows, since the effect of the molecular thermal diffusivity (µ/Pr) becomes verysmall as illustrated in Figure 9.23, it is then necessary to consider the contributionfrom the turbulent thermal diffusivity inside the sub-layer. (Note that a prescribedconstant turbulent Prandtl number Prt is assumed in Figure 9.23.)

In order to compensate the thermal diffusivity inside the sub-layer, Gerasimov[17] introduced an ad hoc effective molecular Prandtl number as:

Preff = Pr

1 + 0.017Pr(1 + 2.9|Fε − 1|)1.5 (74)

where Fε is a model function. This effective Pr approach was tailored for waterflows. Thus, its performance in oil flows whose Pr is over 100 is not guaranteed.

In order to improve the µt profile inside the sub-layer, it is assumed that theprofile of Equation (16) is connected to a functionα′y∗3 at the point y∗

b , as illustratedin Figure 9.23.

µt/µ =α′y∗3, for 0 ≤ y∗ ≤ y∗

bα(y∗ − y∗

v ), for y∗b ≤ y∗ (75)

Thus,

θ =

⎧⎪⎪⎨⎪⎪⎩1 + α′Pry∗3

Prt= θa, for 0 ≤ y∗ ≤ y∗

b

1 + αPr(y∗ − y∗v )

Prt= θb, for y∗

b ≤ y∗(76)

where µθ/Pr is the total thermal diffusivity. By referring to the near-wall profileof µt in a DNS dataset, the value of y∗

b is optimised as y∗b = 11.7 and thus α′ is



obtainable as:

α′ = α(y∗b − y∗

v )

y∗3b

= α

y∗3b

(77)

Using equation (76), integration in equation (18) can be made. Although themodification of the model is very simple, it makes the analytical integration a littlecumbersome as below.

When yb ≤ yn, with P2 = 1/θa, P′2 = 1/θb, y0 = 0, y1 = yb and y2 = yn,

the coefficients DT and ET of equation (45) and (46) are:

DT = S2(y1) − S2(y0) + S ′2(y2) − S ′

2(y1)P2(y1)

P′2(y1)

(78)

ET = S1(y0) − S1(y1) + S ′1(y1) − S ′

1(y2)(79)

+ S ′2(y1) − S ′

2(y2)P1(y1) − P′1(y1)

P′2(y1)

where P1 = y∗P2, P′1 = y∗P′

2, Si = ∫Pidy∗.

In the case of yn < yb, with y0 = 0, y1 = yn, they are:

DT = S2(y1) − S2(y0) (80)

ET = S1(y0) − S1(y1) (81)

(See Appendix B for the results of the integration of 1/θa etc.)

High Pr AWF for rough wall heat transferSince the AWF heat transfer model presented so far was only validated in air flows,the coefficient Ch needs recalibration in high Pr flows and the obtained polynomialform is:

Ch = max (0, C3Pr3 + C2Pr2 + C1Pr + C0) (82)

C3 = −0.48/h∗ + 0.0013, C2 = 9.90/h∗ − 0.0291

C1 = −72.35/h∗ + 0.3067, C0 = 98.98/h∗ + 0.2103

Since the rough wall AWF modifies y∗v of Equation (38) as:

y∗v = y∗

vs

1 −

(h∗

70

)m= y∗

vs − δv (83)

the turbulent viscosity form of equation (75) changes to:

µt/µ =α′(y∗ + δv)3, for y∗ ≤ y∗

bα(y∗ − y∗

v), for y∗b < y∗ (84)



Then, the thermal diffusivity has the following forms:

θ =

⎧⎪⎪⎪⎨⎪⎪⎪⎩1 + α′Pr(y∗ + δv)3

Pr∞t + Ch max (0, 1 − y∗/h∗)= θc, for y∗ < y∗

b

1 + αPr(y∗ − y∗v )

Pr∞t + Ch max (0, 1 − y∗/h∗)= θd , for y∗

b ≤ y∗(85)

The analytical solutions of energy equations then can be obtained in the four dif-ferent cases illustrated in Figure 9.8. The resultant expressions for qw and AT areof the same form as those presented so far. For cases (a) and (d) of Figure 9.8, DT

and ET have the forms of Equations (78) and (79) with some changes. For case (a),they are P2 = P′

2 = 1/θd , y0 = 0, y1 = h and y2 = yn. For case (d), they areP2 = P′

2 = 1/θc, y0 = 0, y1 = h and y2 = yn.In cases (b) and (c), DT and ET have the following forms:

DT = S2(y1) − S2(y0) + S ′2(y2) − S ′

2(y1)P2(y1)

P′2(y1)

+ S ′′2 (y3)

(86)− S ′

2(y2) P2(y1)P′2(y2)

P′2(y1)P′′

2 (y2)

ET = S1(y0) − S1(y1) + S ′1(y1) − S ′

1(y2) + S ′2(y1) − S ′

2(y2)

× P1(y1) − P′1(y1)

P′2(y1)

+ S ′′1 (y2) − S ′′

1 (y3) + S ′′2 (y2) − S ′′

2 (y3) (87)

×(

P1(y1) − P′1(y1)

P′2(y1)

· P′2(y2)

P′′2 (y2)

+ P′1(y2) − P′′

1 (y2)

P′′2 (y2)

)For case (b), P2 = 1/θc, P′

2 = P′′2 = 1/θd , y0 = 0, y1 = yb, y2 = h, and

y3 = yn. For case (c), P2 = P′2 = 1/θc, P′′

2 = 1/θd , y0 = 0, y1 = h, y2 = yb,and y3 = yn. (See Appendix B for the results of the integration of 1/θc etc.)

Examples of applicationsSmooth wall heat transfer. In order to confirm the effects of the corrected turbu-lent viscosity on the flow fields, Figure 9.24 compares the mean velocity profiles inturbulent channel flows at the bulk Reynolds number, Re = 105. (The standard highReynolds number k-ε model [4] and the eddy diffusivity model with Prt = 0.9 areused for the computation of the core fields of the present computations.) Althoughthe result by theµt correction almost perfectly lies on the log-law line and there canbe seen a slight discrepancy between the results with and without the correction,both the results well accord with the LRN LS model and the log-law profiles. (Themeshes used for the AWF and the LRN computations have, respectively, 50 and100 node points in the wall normal direction. Their first cell heights are y+ 30and y+ 0.2, respectively.) This confirms that the correction in the momentumequation may not be totally necessary for engineering flow field computations, andthus the present scheme does not employ the correction for the flow field AWF. This



0

5

10

15

20

25

30

1 10 100 1,000 10,000

U+

y+

Re = 105

LS

k- + AWF(+corr.)k- + AWF(no corr.)

Log − Law

Figure 9.24. Mean velocity profiles in turbulent smooth channel flows.

Θ+

y+

Re = 105Pr = 10.0

LS(Pr = 5.0)Kader

k - ε + AWF(+ corr.)k - ε + AWF(no corr.) 5.0

2.0

0.71

1 100

10

20

30

40

50

60

70

80

100 1,000 10,000

Figure 9.25. Mean temperature profiles in turbulent smooth channel flows.

means that the correction of µt is made only in the energy equation in the presentstrategy.

Figure 9.25 clearly indicates that without the correction, the AWF does notproperly reproduce the logarithmic temperature profiles in high Pr flows of Pr ≥ 5.0.Note that the experimentally suggested logarithmic distribution by Kader [44] fora wide range of Pr is:

+ = 2.12 ln (y+Pr) + (3.85Pr1/3 − 1.3)2 (88)

In the case of Pr = 0.71, the profiles of the AWF with and without the correctionare virtually identical and confirm that the near-wall correction of µt is effectivefor flows at Pr> 1.0.

As shown in Figure 9.26a, the corrected AWF proves its good performance inthe range of 50 ≤ Pr ≤ 103. However, both the LRN LS model and the AWF with



(a)

0

200

400

600

800

1,000

1,200

1,400

1,600

1 10 100 1,000 10,000

Θ+

y +

1 10 100 1,000 10,000

y +

500

100

50

LS (Pr = 500)

k-ε +AWF(Gerasimov,Pr = 500) Kader

(b)

0

2,000

4,000

6,000

8,000

10,000

12,000

14,000

16,000

18,000

Θ+

Re = 105

Re = 105

Pr = 103

Pr = 4 × 104

2 × 104

5 × 103

1 × 104

k-ε +AWF(+corr.)

LS (Pr = 2 × 104)

Kaderk-ε +AWF(+corr.)

Figure 9.26. Mean temperature profiles in turbulent smooth channel flows athigher Pr.

Gerasimov’s [17] effective molecular Prandtl number scheme fail to predict thethermal field at Pr = 500. The former predicts the temperature too high and thelatter does too low. Figure 9.26b also confirms that the corrected AWF performswell up to Pr = 4 × 104 though the LRN LS model predicts the thermal field toohigh. Note that the same grid resolution as that for Pr = 5.0 is used in the LRNcomputations. This reasonably implies that the grid resolution used is too coarseand a much finer grid is needed for such a high Pr computations by the LRN models.Obviously, it highlights the merit of using the AWF which does not require a finergrid resolution for a higher Pr flow.

Rough wall heat transfer. Figure 9.27 compares the predicted temperature fieldsof turbulent rough channel flows of h/D = 0.01 and 0.03, (D is channel height). Inthe cases of h/D = 0.01, the corresponding roughness Reynolds number is h+ 60which is in the transitional roughness regime, while h/D = 0.03 correspondsto h+ 220 which is well in the fully rough regime. For each roughness case,



(a)

0

5

10

15

20

25

30

1 10 100 1,000 10,000

Θ+

y+

Pr = 10.0

5.0

0.71

Pr = 10.05.0

0.712.0

(b)

0

5

10

15

20

25

30

1 10 100 1,000 10,000

Θ+

y+

Re = 105, h/D = 0.01

Re = 105, h/D = 0.03

Kays-Crawfordk-ε+AWF(+corr.)

Kays-Crawfordk-ε + AWF(+corr.)

Figure 9.27. Mean temperature profiles in turbulent rough channel flows.

it is obvious that the corrected AWF reasonably well reproduces the temperaturedistribution for rough walls [45]:

+ = 1

0.8h+−0.2Pr−0.44+ Prt

κln

32.6y+

h+ (89)

where Prt = 0.9 and κ = 0.418. This correlation is based on the experiments ofPr= 1.20 to 5.94 at the order of Re is 104 to 105.

9.4.6 AWF for high Schmidt number flows

Turbulent mass transfer across liquid interfaces is sometimes very important in engi-neering and environmental issues. For example, in order to estimate the amount ofthe absorption of atmospheric CO2 (carbon dioxide) at the sea surface, one shouldconsider predicting turbulent mass transfer across air–water interfaces. The diffi-culty of analysing such a phenomenon comes from that the concentration (scalar)



boundary layer is very much thinner than that of the flow boundary layer. In fact,the Schmidt number Sc is the order of O(103). Since space resolved accurate exper-iments of such free-surface mass transfer are rather difficult compared with thoseof solid walls, numerical approaches are promising (Calmet and Magnaudet [46];Hasegawa and Kasagi [47]) to investigate the detailed mass transfer mechanisms.Although those approaches are based on LES or DNS which require high grid res-olutions for capturing fine-scale flow and scalar transfer, they are not practical tobe used in the estimation of the CO2 absorption at a large water surface even withthe recent computer environment. In order to provide a more practical strategy, theAWF scheme, which requires only rather coarse wall-function grids instead of finegrids resolving the thin scalar boundary layers, can be applied [13].

AWF modelling for high Schmidt number flowsThe theoretical limiting variations of the velocity components and their fluctuationsnear an air–liquid interface are:

u = au + buy + cuy2 + · · · , v = av + bvy + cvy2 + · · ·u′ = a′

u + b′uy + c′

uy2 + · · · , v′ = a′v + b′

vy + c′vy

2 + · · · (90)

Since the eddy viscosity relates the Reynolds stress with the mean velocity gradientas −ρuv = µt∂U/∂y, the above relations lead to:

−ρa′ua′v + (a′

ub′v + a′

vb′u)y + · · · = µt(bu + 2cuy + · · · ) (91)

and it is obvious that µt ∝ O(y0). Thus, the near-interface variation of µt ismodelled as:

µt = αβµmax0, (y∗ − y∗v ) (92)

whereβ is a factor for adjusting the length scale distribution for the near-free-surfaceturbulence. In the case of non-surface disturbance where a′

v = 0, equation (92)

(a)

P Nn

y*(b)

y*yc*

mt abmy*Γt

aby *Sct

∞

Sct∞

acy*2

Figure 9.28. (a) Near surface cell arrangement and the eddy viscosity distribution,(b) turbulent diffusivity distribution.



reduces to µt = αβµy∗ as in Figure 9.28a since equation (91) leads to µt ∝ O(y).On constant surface concentration conditions, the surface limiting behaviour ofconcentration and its fluctuation are:

c = ac + bcy + ccy2 + · · · , c′ = a′c + b′

cy + c′c y2 + · · · (93)

When the turbulent concentration flux −ρvc is modelled as −ρvc = µt∂c/∂y,

−ρa′ca′v+ (b′

ca′v+a′

cb′v)y + (a′

cc′v+c′

ca′v)y

2 +· · · = µt(bc +2ccy +· · · ) (94)

Thus, t ∝ O(y0). (In the case of non-surface-disturbance, t ∝ O(y2) due toa′v= 0 and a′

c = 0.) In the context of the eddy viscosity models, the turbulent scalardiffusivity t is modelled using a turbulent Schmidt number as t = µt/(µSct);thus, with equation (92) one can rewrite this as:

t = αβmax0, (y∗ − y∗v )/Sct (95)

In order to satisfy the limiting behaviour of t near the surface, the limitingbehaviour of Sct is required as O(y−1). Thus, Sct is:

Sct =

Sc∞t /(y

∗/y∗c ), 0 ≤ y∗ ≤ y∗

cSc∞

t , y∗c < y∗ (96)

where Sc∞t is a prescribed constant. This simple two-segment variation profile of

Sct leads to the turbulent diffusivity distribution as in Figure 9.28b:

t =αcy∗2/Sc∞

t , 0 ≤ y∗ ≤ y∗c

αβy∗/Sc∞t , y∗

c < y∗ (97)

for the case of non-surface disturbance where t ∝ O(y2). The coefficient αc andy∗

c have the relation αc = αβ/y∗c for connecting the two segments. Then, with

the assumption that the rhs terms can be constant over the cell, the simplifiedconcentration equation in the surface adjacent cell:

∂

∂y∗

[( µSc

+ µt

) ∂c∂y∗

]= ν2

kP

[∂

∂x(ρUc)

]︸︷︷︸

CC

(98)

can be easily integrated analytically to form the boundary conditions of theconcentration at the surface, namely the surface concentration flux:

qs = − µSc

dc

dy

∣∣∣∣s= −k1/2

p

νAC (99)

or the surface concentration:

cs = cn + qsScDC

ρk1/2P

+ ScCCEC

µ(100)



d

x

y

AirFree surface

Computational domain

Water

Free slip

Figure 9.29. Computational domain of free-surface flows.

where the integration constants AC , DC and EC are

AC = µ(cn − cs)/Sc + CCEC /DC (101)

DC = 1

α1/2cd

tan−1 (α1/2cd y∗

c ) − 1

αctln

(1 + αcty∗

c

1 + αcty∗n

)(102)

EC = 1

αct(y∗

c − y∗n) − 1

α2ct

ln(

1 + αcty∗c

1 + αcty∗n

)− 1

2αcdln

(1 + αcty

∗c

)(103)

with αct = αβSc/Sct and αcd = αct/y∗c . The model coefficients are Sc∞

t = 0.9,β= 0.55 and y∗

c = 11.7.

Examples of applicationsThe example case is a fully developed counter-current air–water flow driven by aconstant pressure gradient. Figure 9.29 illustrates the coordinate system and thecomputational domain applied in the present computations. Only the water phaseis solved with the non-disturbance and the constant concentration conditions at thefree surface. At the bottom of the domain, a free slip and a constant concentration-flux condition are imposed. The coupled turbulence model for core flows is thestandard k-εmodel. Figure 9.30a–c compare the present application results with theDNS results of the free-surface turbulent flow with surface-shear whose Reynoldsnumber is Reτ = 150 by Hasegawa and Kasagi [47]. Figure 9.30a shows that theagreement between the present and the DNS results of the mean velocity distributionis satisfactory and the profiles are significantly lower than that of the log-law linefor wall turbulence. Figure 9.30b and c compare the mean concentration fields of1 ≤ Sc ≤ 103. The concentration profiles reasonably agree with the DNS resultsin the various Schmidt numbers. This means that the AWF model can reproduceessential near-surface behaviour of high Schmidt number turbulent flows.



500

300

100

Sc =1,000

C+ C+

2 5

30

35

20

15

10

5

0

2 50

300

350

400

200

150

100

50

0

(b) (c)

1 10 100

y+1 10 100

y+

PresentDN S

PresentDN S

(a)

2 5

20

15

U+

U+= y+

U+= 2.5 ln y+ + 5.5

10

5

01 10 100

y+

PresentDN S

Figure 9.30. (a) Mean velocity distribution of a turbulent free-shear boundary layer,(b) mean concentration distribution near a free surface at Sc = 1,(c) mean concentration distribution near a free surface at Sc = 100–1,000.

9.5 Conclusions

The AWFs, which are for the replacement of the LWF and the treatment of complexsurface flow phenomena, have been summarised. Their overall strategies for deriva-tion and implementation have been presented with typical application results. Fromthe results, it has been shown that the AWFs are valid in many important flows overa wide range of Reynolds, Prandtl and Schmidt numbers. The effects of wall rough-ness and/or wall permeability can be also included in the AWF as well. Althoughthey do not include the conventional log-laws, the AWFs accurately reproduce thelogarithmic profiles of mean velocities and temperatures without grid dependency.The AWFs perform well with second moment closures, linear and nonlinear k − εmodels while the predictive accuracy depends on the turbulence model employedin the main flow field. The computational cost required for the AWFs is comparableto that of the standard LWF.

In the rough-wall version of theAWF, the effects of wall roughness on turbulenceand heat transfer are considered by employing functional forms of equivalent sandgrain roughness for the non-dimensional thickness of the modelled viscous sub-layer and the turbulent Prandtl number immediately adjacent to the wall.



Although the permeable-wall version of the AWF does not need to resolvethe region inside a permeable wall, by non-linearly linking to the solution of themodified Brinkman equations for the flows inside porous media, it can bridgethe flows inside and outside of the porous media successfully. Since the com-puted turbulent flow fields over permeable porous surfaces are reasonably in accordwith the corresponding data, the AWF for permeable wall turbulence models suchcomplicated wall-flow interactions.

By linking to the correct near-wall variation of turbulent viscosity µt ∝ y3,the AWF can be successfully applicable to flows over a wide range of Prandtlnumbers up to Pr = 4 × 104, for smooth wall cases. (For flow fields and thermalfields at Pr ≤ 1, it is not totally necessary to employ the correct near-wall variationof turbulent viscosity.) For rough wall cases, it is confirmed that the model functionof Prt inside roughness elements enables the AWF to perform well in high Pr flowsat least at Pr ≤ 10.

With a slight modification, the AWF can reproduce the characteristics of theturbulence near free surfaces. The predictive performance of such a version of theAWF is reasonable in turbulent concentration fields in the range of Sc = 1 to 1,000.

Acknowledgements

The author thanks Professors B. E. Launder, H. Iacovides and Dr. T. J. Craft of theUniversity of Manchester for their fruitful discussions and comments on the AWFs.

Appendix A

The analytical integration of the momentum and energy equations (17) and (18) isperformed in the four different cases illustrated in figure 9.8. The process for thecase (a) yv < 0 is as follows:

The integration of equation (17) with equation (40) is:If y∗ ≤ h∗,

dU

dy∗ = AU

µ1 + α(y∗ − y∗v )

= AU

µY ∗ (A.1)

U = AU

αµln Y ∗ + BU (A.2)

At a wall: y∗ = 0, the condition U = 0 leads to:

BU = −AU

αµln (1 − αy∗

v ) = −AU

αµln Y ∗

w (A.3)

When h∗< y∗, the integration leads to:

dU

dy∗ = CU y∗ + A′U

µY ∗ (A.4)



Table 9.A1. Cell averaged generation Pk , and integration constant: AU ; (a) yv <0, (b) 0 ≤ yv ≤ h, (c) h ≤ yv ≤ yn, (d) yn ≤ yv; Uw = 0 fornon-permeable walls

Case Pk

(a)α

µ2y∗n

kP

ν

∫ h∗

0(y∗ − y∗

v )(

AU

Y ∗

)2

dy∗ +∫ y∗

n

h∗(y∗ − y∗

v )(

CU (y∗ − h∗) + AU

Y ∗

)2

dy∗

(b)α

µ2y∗n

kP

ν

∫ h∗

y∗v

(y∗ − y∗v )

(AU

Y ∗

)2

dy∗ +∫ y∗

n

h∗(y∗ − y∗

v )(

CU (y∗ − h∗) + AU

Y ∗

)2

dy∗

(c)α

µ2y∗n

kP

ν

∫ y∗n

y∗v

(y∗ − y∗v )

(CU (y∗ − h∗) + AU

Y ∗

)2

dy∗

(d) 0

Case AU

(a)αµ(Un − Uw) − CU (y∗

n − h∗) + CU

(Y ∗

w

α+ h∗

)ln [Y ∗

n /Y∗h ]

/ln [Y ∗

n /Y∗w ]

(b)αµ(Un − Uw) − CU (y∗

n − h∗) + CU

(Y ∗

w

α+ h∗

)ln [Y ∗

n /Y∗h ]

/(αy∗v + ln Y ∗

n

)(c)

αµ(Un − Uw) − CU (y∗

n − y∗v ) + CU

(Y ∗

w

α+ h∗

)ln Y ∗

n − α

2CU (y∗2

v − 2h∗y∗v + h∗2)

/(αy∗v + ln Y ∗

n

)(d)

µ(Un − Uw) − 1

2CU (y∗2

n − 2h∗y∗n + h∗2)

/y∗

n

U = CU

αµy∗ +

(A′

U

αµ− Y ∗

wCU

α2µ

)ln Y ∗ + B′

U (A.5)

Then, a monotonic distribution condition of dU/dy∗ and U at y∗ = h∗ gives:

AU

µY ∗h

= CU h∗ + A′U

µY ∗h

(A.6)

AU

αµln Y ∗

h + BU = CU

αµh∗ +

(A′

U

αµ− Y ∗

wCU

α2µ

)ln Y ∗

h + B′U (A.7)

Since at y∗ = y∗n the velocity is U = Un (Un is obtainable by interpolating the

calculated node values at P, N ),

Un = CU

αµy∗

n +(

A′U

αµ− Y ∗

wCU

α2µ

)ln Y ∗

n + B′U (A.8)

The integration constants AU , BU , A′U and B′

U are thus easily obtainable by solvingequations (A.3, A.6–A.8).



Table 9.A2. Coefficients in equation (45)

Case DT

(a)−βT h∗αT − βT

+ αT Y βTh

(αT − βT )2ln (Y αT

h /λb) + 1αT

ln (Y αTn /Y αT

h )

(b)αT y∗

v − βT h∗αT − βT

+ αT Y βTh


h /Y βTh ) + 1

αTln (Y αT

n /Y αTh )

(c)1

αTln Y αT

n + y∗v

(d) y∗n

Case ET

(a)h∗ − y∗

n

αT+ 1αT

Y αT

w

αT+ h∗ − YαT

h

αT − βT

(λbβT

αT − βT+ 1

)+ λbαT Y βT

h

(αT − βT )2

ln (Y αT

n /Y αTh )

+ αT λbY βTh


h /λb) − h∗αT − βT

(1 + βT h∗

2+ βT λb

αT − βT

)

(b)h∗ − y∗

n

αT+ 1αT

Y αT

w

αT+ h∗ − YαT

h

αT − βT

(λbβT

αT − βT+ 1

)+ λbαT Y βT

h

(αT − βT )2

ln (Y αT

n /Y αTh )

+ αT λbY βTh


h /Y βTh ) − h∗

αT − βT

(1 + βT h∗

2+ βT λb

αT − βT

)+ y∗

v

αT − βT

(αT Y βT

h

αT − βT− αT y∗

v

2

)

(c)y∗v − y∗

n

αT− y∗2

v /2 + Y αTw

α2T

ln Y αTn

(d) −y∗2n /2

In this case, the wall shear stress τw is written as:

τw = (µ+ µt)dU

dy

∣∣∣∣y=0

= µY ∗w

k1/2P

ν

dU

dy∗

∣∣∣∣y∗=0

= k1/2P AU

ν(A.9)

whose resultant form is the same as equation (27) and can be calculated with AU .Since the cell averaged production term Pk of the turbulence energy k is

written as:

Pk = 1yn

∫ h

0νt

(dU

dy

)2

dy +∫ yn

hνt

(dU

dy

)2

dy

(A.10)

= α

y∗n

kP

ν

∫ h∗

0(y∗ − y∗

v )(

dU

dy∗

)2

dy∗ +∫ y∗

n

h∗(y∗ − y∗

v )(

dU

dy∗

)2

dy∗

it can be calculated by substituting dU/dy∗ with equations (A.1) and (A.4).As for the thermal field, when y∗ ≤ h∗, applying equations (41) and (42) to the

energy equation (18), one can lead to:

∂

∂y∗

[µ

Pr+ αµ(y∗ − y∗

v )Pr∞t + C0(1 − y∗/h∗)

∂

∂y∗

]= CT (A.11)



Its integration follows:

µ

Pr+ αµ(y∗ − y∗

v )Pr∞t + C0(1 − y∗/h∗)

d

dy∗ = CT y∗ + AT (A.12)

d

dy∗ = 1 + βT (h∗ − y∗)(CT y∗ + AT )

1 + αT (y∗ − y∗v ) + βT (h∗ − y∗)

Pr

µ(A.13)

The further integration leads to:

= −

CTλbβT

(αT − βT )2 − CT (1 + βT h∗) + βT AT

αT − βT

Pr

µy∗ + PrCTβT

2µ(αT − βT )y∗2

+[

CTλ2bβT

(αT − βT )3 − λbCT (1 + βT h∗) + βT AT (αT − βT )2

+ (1 + βT h∗)AT

αT − βT

]Prµ

ln [(αT − βT )y∗ + λb] + BT (A.14)

When y∗ > h∗

d

dy∗ = Pr(CT y∗ + A′T )

µ1 + αT (y∗ − y∗v )

= Pr(CT y∗ + A′T )

µY αT(A.15)

= PrCT

µαTy∗ + Pr

µ

A′

T

αT− CT Y αT

w

α2T

ln Y αT + B′

T (A.16)

The integration constants AT , BT , A′T and B′

T are readily obtainable by equations(A.13–A.16) imposing the boundary conditions ( |y=0 = w, |y=yn = n) anda monotonic distribution condition of d /dy∗ and at y∗ = h∗ as in the flowfield.

In the case of a constant wall temperature, the wall heat flux qw can becalculated as:

qw = −ρcp

(ν

Pr+ νt

Prt

)k1/2

P

ν

d

dy∗

∣∣∣∣w

= −ρcpk1/2P

µAT (A.17)

whose resultant form is the same as equation (28) and can be calculated with AT .In the other cases (b)–(d), integration can be performed by the similar

manner.



Appendix B

The integrals of the functions are:

∫1θa

dy∗ =∫

1

1 + α′Pry∗3

Prt

dy∗

(B.1)

= a

3

1

2ln

(y∗ + a)2

y∗2 − ay∗ + a2 + √3 tan−1 2y∗ − a

a√

3

∫

y∗

θady∗ =

∫y∗

1 + α′Pry∗3

Prt

dy∗

(B.2)

= a2

3

−1

2ln

(y∗ + a)2

y∗2 − ay∗ + a2 + √3 tan−1 2y∗ − a

a√

3

∫

1

θbdy∗ =

∫1

1 + αPr(y∗ − y∗v )

Prt

dy∗ = 1

αθln |1 + αθ(y∗ − y∗

v )| (B.3)

∫y∗

θbdy∗ =

∫y∗

1 + αPr(y∗ − y∗v )

Prt

dy∗

(B.4)

= y∗

αθ− 1 − αθy∗

v

α2θ

ln |1 + αθ(y∗ − y∗v)|∫

1θd

dy∗ =∫

1

1 + αPr(y∗ − y∗v )

Pr∞t + Ch max (0, 1 − y∗/h∗)

dy∗

= − βT y∗

αT − βT+

βTλb

(αT − βT )2 + 1 + βT h∗

αT − βT

(B.5)

× ln |(αT − βT )y∗ + λb|∫y∗

θddy∗ =

∫y∗

1 + αPr(y∗ − y∗v )

Pr∞t + Ch max (0, 1 − y∗/h∗)

dy∗

= − βT y∗2

2(αT − βT )+

βTλb

(αT − βT )2 + 1 + βT h∗

αT − βT

y∗ (B.6)

−

βTλ2b

(αT − βT )3 + λb(1 + βT h∗)

(αT − βT )2

× ln |(αT − βT )y∗ + λb|



where αθ = αPr/Prt , αT = αPr/Pr∞t , a = (α′Pr/Prt)−1/3

βT =

Ch/(Pr∞t h∗), for y ≤ h0, for h < y

λb = 1 − βT h∗ − αy∗v . (Note that integration constants are neglected in the

results.)∫1

θcdy∗ =

∫1

1 + α′Pr(y∗ + δv)3

Pr∞t + Ch max (0, 1 − y∗/h∗)

dy∗

= ζ

α′T

[−

ηcβT − (ηa − ηb)(1 + βT h∗) − ηb

2(1 + βT [h∗ + ηa])

(y)

− 1 + βT (h∗ + ηa)

2ln |y∗2 + ηby∗ + ηc|

+ 1 + βT (h∗ + ηa) ln |y∗ + ηa|]

(B.7)

∫y∗

θcdy∗ =

∫y∗

1 + α′Pr(y∗ + δv)3

Pr∞t + Ch max (0, 1 − y∗/h∗)

dy∗

= ζ

α′T

[ηc(1 + βT [h∗ + ηa]) − ηb

2(βT [ηaηb − ηc]

+ ηa[1 + βT h∗])(y) + 1

2βT [ηaηb − ηc] (B.8)

+ ηa(1 + βT h∗) ln |y∗2 + ηby∗ + ηc|

− ηa1 + βT (h∗ + ηa) ln |y∗ + ηa|]

where α′T = α′Pr/Pr∞t ,

ξ =

⎧⎪⎨⎪⎩(−q + √

q2 + p3)1/3 + (−q − √q2 + p3)1/3, if q2 + p3 ≥ 0,

2√−p cos

[1

3cos−1

(q

p√−p

)], if q2 + p3 < 0,

(B.9)

p = −βT /(3α′T ), q = 1 + βT (h∗ + δv)/(2α′

T ),ηa = δv − ξ, ηb = 2δv + ξ, ηc = δ2v + δvξ + ξ2 + 3p, ζ = 1/ηa(ηa − ηb) + ηc



and

(y) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

− 2

ηb + 2y∗ , if η2b − 4ηc = 0,

2√−η2

b + 4ηc

tan−1

⎛⎜⎝ ηb + 2y∗√−η2

b + 4ηc

⎞⎟⎠ , if η2b − 4ηc < 0,

2√η2

b − 4ηc

ln

∣∣∣∣∣∣∣ηb + 2y∗ −

√η2

b − 4ηc

ηb + 2y∗ +√η2

b − 4ηc

∣∣∣∣∣∣∣ , if η2b − 4ηc > 0.

(B.10)

9.6 Nomenclature

AC , AT , AU : integration constantsc, c: concentration, mean concentrationcp: specific heat capacity at constant pressurec, cµ: model constantsC0–C3, Ch: model coefficientsCp: pressure coefficientCC : sum of the convection and the diffusion terms of

the concentration equationCT : sum of the convection and the diffusion terms of

the energy equationCU : sum of the convection and the diffusion terms of

the momentum equationdp: mean particle diameterD pipe diameterDC , DT , EC , ET : integration constantsf : friction coefficient of pipe flowsfi: drag forceF , Fij: Forchheimer correction coefficient, Forchheimer

correction tensorh, h∗: roughness height (equivalent sand grain roughness

height), h√

kP/ν

h+: roughness Reynolds number: hUτ /νH : ramp height or channel heightk , kP: turbulence energy, k at node PK , Kij: permeability, permeability tensor: turbulent length scaleL: ramp lengthNu: Nusselt numberP: pressure or cell centre of the wall-adjacent cellPk : production term of k equationPr, Prt , Pr∞t : Prandtl number, turbulent Prandtl numbers



qw: wall heat fluxqs: surface concentration fluxRe, Rex, Reθ: Reynolds numbers: UbD/ν, U0x/ν, Ueθ/ν; θ:

momentum thicknessReK : permeability Reynolds number: UτK1/2/νSc, Sct , Sc∞

t : Schmidt number, turbulent Schmidt numbersSt: Stanton number: Nu/(RePr)Sθ: source term of the energy equationU , U +: mean velocity, U/UτUb, Uτ : bulk velocity, friction velocity:

√τw/ρ

Ud : Darcy velocityUe, U0: free stream velocitiesUn: velocity at the point nUw: slip velocityx: wall-parallel coordinatex′: x/L in the ramp flowy: wall normal coordinate or wall normal distanceyn, yv, yε: cell height, viscous sublayer thickness, characteristic

dissipation lengthy+, y∗: normalised distances: yUτ/ν, y

√kP/ν

y∗b , y∗

c : connection points in two segment diffusivity modelsY ∗, Y ∗

i : 1 +α(y∗ − y∗v ), 1 +α(y∗

i − y∗v ); i = w, n, h; yw = 0; yh = h

Y αT , Y αTi : 1 +αT (y∗ − y∗

v ), 1 +αT (y∗i − y∗

v ); i = w, n, h; yw = 0; yh = hY βT , Y βT

i : 1 +βT (y∗ − y∗v ), 1 +βT (y∗

i − y∗v ); i = w, n, h; yw = 0; yh = h

α: cµc or heat transfer coefficient: qw/( w − inlet)α′,αct ,αcd : α(y∗

b − y∗v )/y

∗3b , αβSc/Sct , αct/y∗

cαT ,αθ: α Pr / Pr∞t , α Pr / Prt

β,βp: model coefficientsβT : C0/(h∗ Pr∞t )U ,θ ,t: normalised total viscosity, thermal diffusivity and turbulent

diffusivityδv: origin shiftx,y: cell widths in the x, y directionϕ: porosityε: dissipation rate of k , +: mean temperature, | − w|(ρcpUτ)/qw

w, n: wall temperature, temperature at the point nκ, κt : von Kármán constantsλb: Y αT

w +βT h∗µ,µt : viscosity, turbulent viscosityν, νt: kinematic viscosity, kinematic turbulent viscosityρ: fluid densityτw: wall shear stress〈φ〉, 〈φ〉f : superficial and intrinsic averaged values of φφ: Reynolds averaged value of φ



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[2] Kim, J., Moin, P., and Moser, R. Turbulence statistics in fully developed channel flowat low Reynolds number, J. Fluid Mech., 177, pp. 133–166, 1987.

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[9] Cabot, W., and Moin, P. Approximate wall boundary conditions in the large-eddysimulation of high Reynolds number flow, Flow Turb. Combust., 63, pp. 269–291,1999.

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[13] Suga, K., and Kubo, M. Modelling turbulent high Schmidt number mass transfer acrossunderformable gas-liquid interfaces, Int. J. Heat Mass Transfer, 53, pp. 2989–2995,2010.

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[19] Suga, K. A study toward improving accuracy of large scale industrial turbulent flowcomputations, in: Frontiers of Computational Science, Y. Kaneda, H. Kawamura, M.Sasai (Eds.), Springer, pp. 99–105, 2007.



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Pub. Co, Washington, 1977.[22] Krogstad, P. A., Antonia, R. A., and Browne, L. W. B. Comparison between rough- and

smooth-wall turbulent boundary layers, J. Fluid Mech., 245, pp. 599–617, 1992.[23] Tachie, M. F., Bergstrom, D. J., and Balachandar, R. Roughness effects in low-Re0

open-channel turbulent boundary layers, Expt. Fluids, 35, pp. 338–346, 2003.[24] Nagano, Y., Hattori, H., and Houra, T. DNS of velocity and thermal fields in turbulent

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[25] Launder, B. E., and Sharma, B. I. Application of the energy-dissipation model ofturbulence to the calculation of flow near a spinning disc, Lett. Heat Mass Transfer, 1,pp. 131–138, 1974.

[26] Craft, T. J., Launder, B. E., and Suga, K. Development and application of a cubiceddy-viscosity model of turbulence, Int. J. Heat Fluid Flow, 17, pp. 108–115, 1996.

[27] Moody, L. F. Friction factors for pipe flow, Trans. ASME, 66, pp. 671–678, 1944.[28] Turner, A. B., Hubbe-Walker, S. E., and Bayley, F. J. Fluid flow and heat transfer

over straight and curved rough surfaces, Int. J. Heat Mass Transfer, 43, pp. 251–262,2000.

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[30] Iacovides, H., and Raisee, M. Recent progress in the computation of flow and heattransfer in internal cooling passages of gas-turbine blades, Int. J. Heat Fluid Flow, 20,pp. 320–328, 1999.

[31] Song, S., and Eaton, J. K. The effects of wall roughness on the separated flow over asmoothly contoured ramp, Expt. Fluids, 33, pp. 38–46, 2002.

[32] Zippe, H. J., and Graf, W. H. Turbulent boundary-layer flow over permeable and non-permeable rough surfaces, J. Hydraul. Res., 21, pp. 51–65, 1983.

[33] Breugem, W. P., Boersma, B. J., and Uittenbogaard, R.E. The influence of wallpermeability on turbulent channel flow, J. Fluid Mech., 562, pp. 35–72, 2006.

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[45] Kays, W. M., and Crawford, M. E. in: Convective Heat and Mass Transfer, 3rd ed.,McGraw-Hill, New York, pp. 298–301, 1993.

[46] Calmet, I., and Magnaudet, J. High-Schmidt number mass transfer through turbulentgas-liquid interfacesInt. J. Heat Fluid Flow, 19, pp. 522–532, 1998.

[47] Hasegawa, Y., and Kasagi, N. Effects of interfacial velocity boundary condition onturbulent mass transfer at high Schmidt numbers, Int. J. Heat Fluid Flow, 28, pp. 1192–1203, 2007.


IV. Advanced Simulation ModelingTechnologies


10 SPH – a versatile multiphysicsmodeling tool

Fangming Jiang1 and Antonio C.M. Sousa2, 3

1The Pennsylvania State University, USA2Universidade de Aveiro, Portugal3University of New Brunswick, Canada

Abstract

In this chapter, the current state-of-the-art and recent advances of a novel numericalmethod – the Smoothed Particle Hydrodynamics (SPHs) will be reviewed throughcase studies with particular emphasis to fluid flow and heat transport. To providesufficient background and to assess its engineering/scientific relevance, three par-ticular case studies will be used to exemplify macro- and nanoscale applications ofthis methodology. The first application deals with magnetohydrodynamic (MHD)turbulence control. Effective control of the transition to turbulence of an electri-cally conductive fluid flow can be achieved by applying a stationary magnetic field,which is not simply aligned along the streamwise or transverse flow direction, butalong a direction that forms an angle with the main fluid flow in the range of 0(streamwise) to 90 (transverse). The SPH numerical technique is used to interpretthis concept and to analyze the magnetic conditions. The second application dealswith non-Fourier ballistic-diffusive heat transfer, which plays a crucial role in thedevelopment of nanotechnology and the operation of submicron- and nanodevices.The ballistic-diffusive equation to heat transport in a thin film is solved numericallyvia the SPH methodology. The third application deals with mesoscopic pore-scalemodel for fluid flow in porous media. SPH simulations enable microscopic visu-alization of fluid flow in porous media as well as the prediction of an importantmacroscopic parameter – the permeability.

Keywords: CFD, Numerical methods, Smoothed particle hydrodynamics

10.1 Introduction

A novel numerical method – the smoothed particle hydrodynamic (SPH) offers arelatively flexible tool for heat and fluid flow computations, as it can cope with awide range of space scales and of physical phenomena. SPH is a meshless particle-based Lagrangian fluid dynamics simulation technique, in which the fluid flow is



represented by a collection of discrete elements or pseudo-particles. These par-ticles are initially distributed with a specified density distribution and evolve intime according to the constitutive conservation equations (e.g., mass, momentum,energy). Flow properties are determined by an interpolation or smoothing of thenearby particle distribution using a special weighting function – the smoothingkernel. SPH was first proposed by Gingold and Monaghan [1] and by Lucy [2]in the context of astrophysical modeling. The method has been successful in abroad spectrum of problems, among others, heat conduction [3, 4], forced andnatural convective flow [5, 6], low Reynolds number flow [7, 8], and interfacialflow [9–11]. Many other applications are given by Monaghan [12] and Randle andLibersky [13], which offer comprehensive reviews on the historical backgroundand some advances of SPH. In comparison to the Eulerian-based computationalfluid dynamics (CFD) methods, SPH is advantageous in what concerns the follow-ing aspects: (a) particular suitability to tackle problems dealing with multiphysics;(b) ease of handling complex free surface and material interface; and (c) relativelysimple computer codes and ease of machine parallelization. These advantages makeit particularly well suited to deal with transient fluid flow and heat transport.

This chapter is organized in six main sections, namely: 1. Introduction; 2. SPHtheory, formulation, and benchmarking; 3. Control of the onset of turbulence inMHD fluid flow; 4. SPH numerical modeling for ballistic-diffusive heat conduc-tion; 5. Mesoscopic pore-scale SPH model for fluid flow in porous media, and 6.Concluding remarks.

10.2 SPH Theory, Formulation, and Benchmarking

SPH, despite its promise, is far from being at its perfected or optimized stage.Problems with the SPH, such as handling of solid boundaries [14], choice of smooth-ing kernel and specification of the smoothing length [15], treatment of heat/massconducting flow [5, 6], and modeling of low Reynolds number flow, where thesurface/interface effects act as a dominating role relative to the inertial effect [10],have prevented its application to some practical problems. This section reportsan effort to enhance the SPH methodology and develop appropriate computerprogrammable formulations.

10.2.1 SPH theory and formulation

In the SPH formulation, the continuous flow at time t is represented by a collec-tion of N particles each located at position ri(t) and moving with velocity vi(t),i = 1, 2, . . . . . . , N . The “smoothed” value of any field quantity q(r, v) at a spacepoint r (bold text designates vector or tensor) is a weighted sum of all contributionsfrom neighboring particles, namely:

〈q(r, v)〉 =∑

j

mj

ρ(rj)q(rj, vj) w (|r − rj|, h) (1)


SPH – a versatile multiphysics modeling tool 385

〈∇q(r, v)〉 =∑j =i

mj

ρ(rj)q(rj, vj)∇w(|r − rj|, h) (2)

where, mj and ρ(rj), respectively, denote the mass and density of particle j. w(|r|, h)is the weight or smoothing function with h being the smoothing length.

Density updateBased on equation (1), the local density at position ri(t) is:

ρi =∑

j

mjwij (3)

The summation is over all neighboring particles j including particle i itself. Equa-tion (1) conserves mass accurately if the total SPH particles are kept constant. Interms of equation (2), another formulation for density calculation can be derivedfrom the continuity equation, namely

dρi

dt=

∑j =i

mjvij · ∇iwij (4)

The summation is over all neighboring particles j with exception of particle i itself.Here, vij = vi − vj, rij = ri − rj, and wij = w(|ri − rj|, h); the subscript in the oper-ator ∇i indicates the gradient derivatives are taken with respect to the coordinatesat particle i

∇iwij = rij∣∣rij∣∣ ∂wij

∂ri(5)

Equation (4) is the Galilean invariant, which is more appropriate for interfacereconstruction in free surface flow simulations. The use of this equation has thecomputational advantage over equation (3) of calculating all rates of change inone pass over the particles; whereas the use of equation (3) requires one pass tocalculate the density, then another one to calculate the rates of change for velocity,and different scalars, such as temperature.

Equation (4) is employed to perform the density update in the case studies tobe presented.

IncompressibilityA quasi-compressible method is implemented to determine the dynamic pressure p[8], in which an artificial equation of state is used

p = p0

[(ρ

ρ0

)γ− 1.0

](6)

where, p0 is the magnitude of the pressure for material state corresponding to thereference density ρ0. The ratio of the specific heats γ is artificially a large value



(say, 7.0) to guarantee the incompressibility of the fluid. The speed of sound, cs,can be formulated as:

c2s = γp0

ρ0= αU 2 (7)

where, U is the characteristic or maximum fluid velocity. The choice of α is acompromise: it should be an adequate value to avoid making p0 (or cs) too large,which will force the time advancement of the simulation to become prohibitivelyslow, and also to avoid yielding Mach numbers that can violate the incompressibilitycondition of the fluid; α takes 100 in the case studies to be presented, which ensuresthe density variation be less than 1.0%.

Velocity rate of changeThe SPH formulation of viscous effects in fluid flow requires careful modeling tosatisfy its adaptivity and robustness. In this respect, Morris et al. [8] presented thefollowing expression, equation (8), to simulate viscous diffusion.(

1ρ

∇ · µ∇)

v

i=

∑j =i

mj(µi + µj)vij

ρiρj

(1

|rij|∂wij

∂ri

)(8)

This expression has proven to be highly accurate in particular for the simulations oflow Reynolds number planar shear flows [7, 8]. The variableµ denotes the dynamicviscosity of the fluid.

In what concerns equation (2), the pressure gradient terms in the Navier–Stokes (N–S) equations can be formulated using the following symmetrized [6]SPH version:

−(

1ρ

∇p

)i= −

∑j =i

mj

(pi

ρ2i

+ pj

ρ2j

)∇iwij (9)

Therefore, the SPH momentum equation yields

dvi

dt= −

∑j =i

mj

⎡⎢⎢⎢⎣(

pj

ρ2j

+ pi

ρ2i

)∇iwij − 1.0

ρiρj(µi + µj

) ∇iwij · rij

r2ij + 0.01h2

vij

⎤⎥⎥⎥⎦ + Fi (10)

where, F is the body force per unit mass (m/s2). The quantity 0.01 h2 is used toprevent singularities for rij ≈ 0.

Position updateTo assure the SPH particles move with a velocity consistent with the average velocityof its neighboring particles it used the variant proposed by Morris [10]

dri

dt= vi + ε

∑j

mjvij

ρijwij (11)

Here, ρij = (ρi + ρj)/2.0 and the factor ε is chosen as ε= 0.5.



Rate of change of temperatureCleary et al. [16] derived a SPH heat transfer equation for solidifying metals, whichcan be rewritten for a fluid of constant heat capacity as:

4.0mj

ρiρj

λiλj

λi + λj

rij · ∇iwij

r2ij + 0.01h2

Tij −

2.0mj

ρiρj

µiµj(vij · rij

)2

µi + µj· rij · ∇iwij(

r2ij + 0.01h2

)2

cpdTi

dt=

∑j =i

(12)

where, λ and cp are the heat conductivity and the specific heat at constant pressure,respectively. Due to the incompressibility of the fluid, pressure work was neglectedand the second term on the right-hand side denotes the contribution from viscousheat dissipation.

Time advancementThe 4th order Runge–Kutta integration algorithm is used to perform the integrationsof equations (4), (10), (11), and (12). The time step t is adaptive and determinedby the Courant–Friedrichs–Lewy (CFL) condition [8] given by:

t = min

⎛⎝ c1h

maxi(15.0U , |vi|) , c2

√h

max acceleration,

c3h

max sound speed

⎞⎠ (13)

The Courant “safety” factors c1, c2, and c3 are given as c1 = 0.125, c2 = 0.25, andc3 = 0.25, respectively. In the right-hand side of equation (13), a quantity 15.0Uis introduced in the first constraint so that the three constraints may have similarinfluence upon the calculation.

Smoothing kernelHigh accuracy three-splines quintic kernel is used

wij = nd1.0

hβ

⎧⎪⎪⎨⎪⎪⎩(3.0 − s)5 − 6.0 (2.0 − s)5 + 15.0 (1.0 − s)5 if 0 ≤ s < 1.0(3.0 − s)5 − 6.0 (2.0 − s)5 if 1.0 ≤ s < 2.0(3.0 − s)5 if 2.0 ≤ s < 3.00 if s ≥ 3.0

(14)

where, s = |rij|/h, β is the dimensionality of the problem, and nd a normalizationconstant, which is determined by the normalization property [15] of the smoothingkernel ∫

w(|r − r′|, h)dr′ = 1.0 (15)

For the present implementation, β= 2 and nd = 7.0/478.0π.



BoundarydA

dB

vA

vB B

Af

x

y

Fluid channel

Wall region ∆p

∆p

∆p /2∆p /2

Figure 10.1. Illustration of initial particle arrangement and treatment of solid wallboundary.

The smoothing length h is given as:

h = 1.001 ×√(x)2 + (y)2 (16)

where, x and y are the initial spacing in the x and y direction, respectively,between two neighboring particles, and x =y. Thus, the interpolation of thefield quantity with respect to one fluid particle extends to a region where about 61neighboring particles are included.

Solid wall boundaryThe treatment of solid wall boundaries is illustrated in Figure 10.1. Taking intoaccount the influence region of the employed quintic smoothing kernel, four layersof ‘dumb’wall particles are arranged outside of the fluid channel and parallel to thereal wall boundary with the first layerp/2 away from the wall plane. These ‘dumb’particles have the same spatial separation p as fluid particles. The density of a‘dumb’particle is initially set equal to the reference fluid density, ρ0, and it remainsunchanged. Boundary particles interact with the fluid particles by contributingto their density variations, and by prescribing viscous forces on the nearby fluidparticles.

To ensure the non-penetrating condition of solid wall boundary, the calculationsare made with boundary particles exerting central forces on fluid particles [17]. Fora pair of boundary and fluid particle separated by a distance r, the force per unitmass f (r) is assumed to have the 12-6 Lennard–Jones form [18]:

f (r) = D

[( r0

r

)12 −( r0

r

)6]

r

r2 (17)

f (r) is set to zero if r> r0 so that the force is purely repulsive. r0 is taken to be0.45p. During the calculations, only the component of f (r) normal to the solidwall is considered; D has the units of (m/s)2 and is taken as 10.0F × Characteristic-Geometric-Length. Varying the magnitude of D has no particular influence on thecalculation results.

Nonslip wall boundary condition is guaranteed by using the method proposedby Morris et al. [8]. For each fluid particle A, the normal distance dA to the wall is



y

d

ux (y, ∞)

U0

xB0 B0B 0

−d

0

Figure 10.2. Combined Poiseuille and Couette flow.

calculated and for each boundary particle B, dB is obtained. Then an artificial veloc-ity vB = −(dB/dA)vA is applied to the boundary particle B with an assumption ofzero velocity condition on the boundary plane itself. This artificial velocity vB isused to calculate the viscous force applied on the nearby fluid particles, but notused to locate the boundary particle position. Boundary particles are motionlessor moving at specified velocities. The relative velocity between fluid and boundaryparticles is:

vAB = vA − vB = χvA (18)where

χ = min(χmax, 1.0 + dB

dA

)(19)

χmax is used to exclude the extremely large values when fluid particles get tooclose to the wall. In Morris et al. [8] good results were obtained for χmax = 1.5when simulating low Reynolds number planar shear flows. In the present study,χmax = 7.0 is specified, which is the largest value of χ for the initial regular particlearrangement.

10.2.2 Benchmarking

The validity of the above-outlined SPH formulation is examined for two bench-marks dealing with heat and fluid flow, namely: combined transient Poiseuille andCouette flow, and steady heat convection for combined Poiseuille and Couette flowconditions; these benchmarks were selected because analytical solutions for themare available.

Transient combined Poiseuille and Couette flowThis test case is the fluid flow between two parallel infinite plates, located at y = −dand y = d, respectively, as shown in Figure 10.2. The system is initially at rest. Attime t = 0, the upper plate starts to move at constant velocity U0 along the positivex-axis. Concurrently, a body force F is aligned parallel to the x-axis. The resultantvelocity profile is dependent on a non-dimensional quantity, designated as B

B = d2

νU0F (20)



and its time-dependent series solution is:

ux(y, t) = F

2ν(d2 − y2) + U0

2d(y + d)

−∞∑

n=0

(−1)n 16Fd2

νπ3(2n + 1)3 cos(πy

2d(2n + 1)

)exp

(− (2n + 1)2π2ν

4d2 t

)

+∞∑

n=1

(−1)n 2U0

nπsin

(nπ

2d(y + d)

)exp

(−n2π2ν

4d2 t

)(21)

The first and third terms in the right-hand side of equation (21) refer to the contri-bution by the applied body force F and the other two terms result from the motionof the upper plate. When time t approaches infinite, the steady-state solution takesthe form:

ux(y, ∞) = B

2U0

(1 − y2

d2

)+ 1

2U0

(1 + y

d

)(22)

The physical properties of the fluid are specified as, ν= 1.0 × 10−6 m2/s andρ= 1000.0 kg/m3. The value of d is taken as 0.0005 m and U0 5.0 × 10−5 m/s. FiftySPH fluid particles are set to span the channel. The body force F is adjusted to givedifferent values of B. The results are presented in Figures 10.3a and b.

The overall agreement between simulations and exact analytical solutions isexcellent: the largest deviation between SPH simulation and series solution isobserved at the position close to the motionless wall, and it is less than 2.0%.

Steady heat convection for combined Poiseuille and Couette flow conditionsThe second test case is conducted on the steady convective heat transfer betweentwo parallel infinite plates, which are kept at constant temperatures: Te and Tw(Te> Tw), respectively. The upper plate is moving at a constant velocity U0 yieldingthe Couette flow condition, as shown in Figure 10.4. Concurrently, a body force Fmay be applied too.

The exact solution for the steady-state temperature profile is given as:

T ∗ (y) = 12

(1 + y∗) + Br

8(1 − y∗2) − Br

6B(y∗ − y∗3) + Br

12B2(1 − y∗4) (23)

where, Br stands for the Brinkman number which is the product of the Prandtlnumber Pr by the Eckert number Ec (Br = Pr.Ec), y∗ and T ∗ are the normalizedposition and temperature, respectively, as defined below:

Br = µcp

λ

U 20

cp(Te − Tw)(24)



4.8

(a)

4.0

3.2

2.4

1.6

0.8

0.0

−1.0

−1.0

−0.5 0.0

2.0

5.0

B = 10.0

0

y /d

u/U

0

0.5 1.0

4.8

5.6

(b)

4.0

3.2

2.4

1.6

0.8

0.0

−1.0

−1.0

−0.5 0.0

2.0

5.0

B = 10.0

0

y /d

u/U

0

0.5 1.0

Figure 10.3. Results for combined Poiseuille and Couette flow at: (a) time =0.212 seconds, and (b) time approaching infinite. (Symbols representthe SPH predictions and the full lines the analytical solutions).

y∗ = y

d(25)

T ∗(y) = T (y) − Tw

Te − Tw(26)

The parameter B is the ratio between the contributions from the body force to thefluid flow and the motion of upper plate, as expressed in equation (20). The first termin the right-hand side of equation (23) is due to the heat conduction between the two



y

du U0, T Te

u 0, T Tw

u ux (y)

U0

x

−d

0Body force FT T(y)

Figure 10.4. Steady heat convection for combined Poiseuille and Couette flowcondition.

parallel plates, whereas the last three terms describe the viscous heat dissipation,which for moderate values of B and small values of Br is negligible; however, forhigh viscosity fluids, as expected, may be significant. In the following calculations,to test the ability of the SPH methodology in predicting heat dissipation, the fluidis assumed to have a relatively large viscosity of 1.0 Pa·s. Moreover, the geometricdimension is increased to d = 0.05 m and the moving velocity of the upper plate isU0 = 0.05 m/s. Other parameters specified are: heat conductivity, λ= 0.1W/m·K;heat capacity, cp = 4,200 J/kg·K; density, ρ= 1,000.0 kg/m3; and the temperatureof the lower plate, Tw = 0C. By varying the magnitude of F and the upper platetemperature Te, different B and Br values are obtained.

As before, 50 SPH fluid particles are set to span the channel. The results forthis particular case are depicted in Figures 10.5a and b, and, as observed, the SPHpredictions for the non-dimensional temperature compare well with the analyticalsolution; the maximum deviation is less than 1%.

The heat convection process and the heat dissipation characteristics are alsoreproduced well by the SPH simulations.

10.3 Control of the Onset of Turbulence in MHD Fluid Flow

Magnetohydrodynamic (MHD) flow control finds growing importance in manyindustrial settings, in particular in the metallurgical industry. The electromagneticforce imposed by a stationary magnetic field may suppress the flow instabilitiesof electrically conducting fluids and creates favorable flow conditions in fluid flowdevices, such as tundishes, as described by Szekley and Ilegbusi [19]. The workreported by Hartmann and Lazarus [20] was the first one to experimentally identifythe change in drag and the suppression of turbulence when a magnetic field is appliedto turbulent liquid metal flows. From then on, numerous studies have addressed theinteractions between the applied transverse (wall-normal) magnetic field and thefluid flow, as reported in Lee and Choi [21], Krasnov et al. [22], Ji and Gardner [23].A magnetic field applied in the transverse flow direction is effective in reducingthe turbulence fluctuations and suppressing the near-wall streamwise vorticity; themean velocity profile is modified, therefore yielding a drag increase. In contrast to



(a)

1.2

1.6

0.8

T*

T*

0.0

0.4

−1.0 −0.5 0.0

2.0

5.0

Br = 0.001

10.0

y /d0.5 1.0

(b)

6.0

8.0

10.0

4.0

0.0

2.0

−1.0 −0.5 0.0

0.4

Br = 0.0001

1.0

y /d0.5 1.0

Figure 10.5. Results for temperature profiles when (a) B = 0 and (b) B = 10.0.(Symbols represent the SPH predictions and the full lines theanalytical solutions).

the extensive research effort dealing with MHD flows with an applied transversemagnetic field, little attention was given to the flows interacting with a streamwisemagnetic field. The application of a streamwise magnetic field, to a flow, however,is able to restrain the velocity fluctuations in the transverse plane of that flow andthe transition to turbulence can be distinctly delayed, or even suppressed in the flow.The ability of a streamwise magnetic field of controlling the onset of turbulencein an electrically conducting fluid flow was recently corroborated by Jiang et al.[24]. To achieve the most effective flow control with the least amount of energyconsumption is the goal of an MHD flow control strategy, which can be realized bycombining the two above-mentioned magnetic fields.



10.3.1 MHD flow control

Motion of electrically conducting fluids across a magnetic field induces the so-calledLorentz force, which, when properly handled, may suppress the flow instabilityand restrain the flow turbulence. A magnetic field applied in the transverse (wall-normal) direction of a fluid flow creates a ponderomotive body force and changesthe mean velocity profile due to the formation of the Hartmann layers at electricallyinsulating boundaries normal to the magnetic field. A magnetic field applied to theflow in the streamwise direction restrains the velocity fluctuations in the transverseplane of the fluid flow and prevents the onset of turbulence. In comparison toa streamwise magnetic field, a magnetic field applied in the transverse directionof the flow acts directly on the main flow and may prove to be more flexible incontrolling the flow; however, it increases the flow resistance and needs additionalenergy consumption. Therefore, a reasonable control strategy may be realized bybringing an angle between the magnetic field and the flow direction within therange of the two above-mentioned magnetic supply conditions, i.e., 0 (streamwise)and 90 (transverse). In this way, both velocity fluctuations in the transverse andstreamwise direction are restrained concurrently and a more effective control effectmay be achieved with some advantage in what concerns external energy usage.

The transition to turbulence of electrically conducting fluid flow in a two-dimensional (2-D) channel constrained by two parallel infinite plates, which areassumed electrically insulated, as depicted in Figure 10.6, is chosen to illustrate theunderlying concept. Fluid enters the channel with a uniform velocity U0, initially.

For a Reynolds number exceeding a critical value (≈2,000), the turbulence, aftera short entry length, sets in. For the sake of restraining the transition to turbulence, auniform magnetic field B is applied at an angle θ to the main fluid flow direction.Thepresent study, by adjusting the angle θ, aims to achieve the best restraining effect. Tocompare the results obtained for different values of θ, a body force FB is prescribedin the x direction to balance the produced ponderomotive electromagnetic forceso as to maintain the maximum flow velocity ( ≈U0) approximately equal for thecases of interest. The restraining effect is quantitatively evaluated by the time whenthe turbulence sets in, or by measuring the magnitude of the “turbulence-related”quantities – the mean velocity u and the two velocity correlations in the x and ydirection, i.e., u′u′ and v′v′ at a specified time.

B

θ

FBU0 u (y, t)

y

x

Figure 10.6. MHD flow control for 2-D channel flow.



10.3.2 MHD modeling

The MHD flow is governed by a set of coupled partial differential equations thatexpress the conservation of mass, momentum and the interaction between the flowand the applied magnetic field. In the present work, it is assumed that the hydro-dynamic Reynolds number Re is much larger than the magnetic Reynolds numberRem, namely

Re = UL

ν>>Rem = UmLm

η(27)

Here, η= (σµ0)−1, where ν, σ, and µ0 are the kinematic viscosity, electricalconductivity, and magnetic permeability of free space (µ0 = 4π× 10−7 H/m),respectively; U and L are the characteristic velocity and length of the fluid flow,respectively; the subscript m denotes the corresponding magnetic quantities. Thisassumption guarantees the fluctuations of the magnetic field due to the fluid motionare very small as compared to the applied field B; therefore, the total magnetic fieldcan be taken as uniform and time-independent, and the induced electrical current jis given by:

j = σ(∇φ + V × B) (28)

where, V is the fluid velocity vector and φ is the induced electrical potential, whichis produced by the interaction between the applied magnetic field, and the flowvorticity w (=∇ ×V ). The present study does not aim to investigate the turbulencepattern or structure of MHD turbulent flows [21–23], of interest here is to investigatethe time duration prior to the onset of turbulence of the flow. The vorticity itself,w, is considered to be negligible, and the induced electrical potential is assumed tobe zero, which yields [25]

j = σ(V × B) (29)

The Lorentz force defined per unit of volume, F takes the form

F = j × B = σ(V × B) × B (30)

Accordingly, the MHD equations for the above-mentioned transient 2-D (x − y)planar channel flow can be written as follows:

Dρ

Dt= −ρ

(∂u

∂x+ ∂v

∂y

)(31)

ρDu

Dt= −∂p

∂x+ µ

(∂2u

∂x2 + ∂2v

∂y2

)− σ(B sin θ)2u + FB (32)

ρDv

Dt= −∂p

∂y+ µ

(∂2u

∂x2+ ∂2v

∂y2

)− σ(B cos θ)2v (33)



where, u and v denote the velocity components in the streamwise (x) and transverse(y) direction, respectively. In the present study, u is the total velocity in the xdirection: u = u + u′, and v is the velocity fluctuation in the y direction, i.e., v= v′.ρ, and µ stand for the density and dynamic viscosity, respectively. D/Dt is thematerial derivative. FB (N/m3) is the applied body force, which is used to balancethe electromagnetic force in the x direction, and is given by:

FB = σ(B sin θ)2U0 (34)

10.3.3 SPH analysis of magnetic conditions to restrain the transitionto turbulence

SPH simulations are performed with respect to the channel flow configurationshown in Figure 10.6. The separation distance between the two parallel infiniteplates (L) is taken as 1.0 m. The computational domain is a unit square where solidboundaries confine the flow in the transverse direction (y) and periodic boundariesare specified in the flow direction (x). The simulation is started with the fluid movingto the right (positive x) at a velocity, U0 = 1.0 m/s. The Reynolds number is definedas Re = UL/ν, with U , L being the characteristic velocity (1.0 m/s) and length(1.0 m), respectively, and the kinematic viscosity, ν, is adjusted to give differentReynolds numbers. The initial arrangement of the SPH particles and treatment ofsolid wall boundaries were already illustrated in Figure 10.1. A 50 square lattice offluid particles is used.

The intensity of the electromagnetic force can be quantified with a dimensionlessquantity, namely, Stuart number St, which is defined as:

St = Rem Al = σB2Lm

ρUm(35)

where, Al denotes the Alfvén number. The Stuart number St designates the ratioof the electromagnetic force to the inertial force. It is assumed that the magneticcharacteristic velocity Um and length Lm have the same magnitudes of their hydro-dynamic counterparts, i.e., 1.0 m/s for Um and 1.0 m for Lm. The magnitude of theapplied magnetic field B is adjusted to give different values of St.

Transitional organization of the SPH fluid particles versus the onsetof turbulenceFor SPH simulations, the onset of turbulence is accompanied by an ordered–disordered transition of the SPH fluid particles [14, 24], so the transitionalorganization of the SPH fluid particles indicates the time when the onset of tur-bulence takes place. For fluid flow of Re = 104 and St = 40 with the magnetic fieldaligned along the streamwise flow direction, i.e., θ= 0, at the time of 1.89 secondsthe SPH particles display an ordered flow pattern, which takes on a plug flow pro-file with very thin boundary layers. At 1.95 seconds the transitional organization ofSPH fluid particles begins to appear, which indicates that the flow turbulence sets



−0.5

0.0

0.6 0.8 1.0

y/L

0.5

u (m/s)

−0.5

0.0

0.02 0.03 0.04 0.05

y/L

0.5

u ′u ′ ( m 2/ s 2)

y/L

−0.5

0.0

0.0 5.0×10−4 1.0×10−3 1.5×10−3 2.0×10−3

0.5

ν′ν′ ( m 2/ s 2)

Figure 10.7. Representation of u, u′u′, and v′v′ for MHD channel flow of Re = 104,St = 40, and θ= 0 at time ∼1.9 seconds.

in. The calculation is terminated at this point because the ensuing MHD turbulenceis of no immediate interest to the present study.

Further insight into the flow state can be achieved by examining the turbulence-related quantities, the mean velocity u and the velocity correlations u′u′ and v′v′.This is done by dividing the flow domain into 50 horizontal strips and by usinga smaller time step to redo the calculations around a specified instant of time,afterwards the results are averaged both spatially and temporally over each sub-region. Shortly before the onset of turbulence is detected, at time ∼1.9 seconds,the turbulence-related quantities of the above-described MHD fluid flow are shownin Figure 10.7. The flow takes on some turbulent flow characteristics, namely, themean velocity profile is approaching to that of plug flow with the occurrence ofvery thin boundary layers.

Enhanced effect of the tilting angle θ

The MHD flow to be considered in this section has Re = 104 and St = 40. The tiltingangle θ of the applied magnetic field B to the main fluid flow direction is varied toexamine the acquired restraining effect to the transition to turbulence. When θ= 0,i.e., a streamwise magnetic field is applied, the flow turbulence takes place at time∼1.95 seconds. The turbulence-related quantities at time ∼1.9 seconds, as shownin Figure 10.7, reveal the velocity correlations in the y direction take smaller valuesthan those in the x direction: the values of u′u′are in the range of 0.015 to 0.06 m2/s2



0.0 5.0×10−15 1.0×10−14 1.5×10−14

y/L

−0.5

0.0

0.5

u ′u ′ ( m 2/ s 2)

y/L

−0.5

0.0

0.5

0.0 6.0×10−9 1.2×10−8 1.8×10−8

ν′ν′ ( m2/ s2)

0.94 0.96 0.98 1.00

y/L

−0.5

0.0

0.5

u (m/s)

Figure 10.8. Representation of u, u′, and v′v′ for MHD channel flow of Re = 104,St = 40, and θ= 20 at time ∼5.24 seconds.

while v′v′ are within 0 to 0.002 m2/s2, which implies that the velocity fluctuations inthe transverse plane of the flow are restrained by the applied streamwise magneticfield. The largest velocity fluctuation in the y direction is detected to occur in theregion close to the walls, whereas that in the x direction takes place in the centralregion of the fluid flow. This indicates the velocity fluctuation in the x directionis initiated from the large bulk flow velocity, while the one in the y direction iscaused by the interactions between the fluid flow and the boundary walls. It is thelarge velocity fluctuations freely developing in the x direction that cause the velocityfluctuation in the y direction and yield the transitional organization of SPH particles.An effective MHD method to restrain the transition to turbulence is enhanced whenthe applied magnetic field is capable of controlling the flow concurrently in the xand y direction.

At an angle of θ= 20, plug flow with very thin boundary layers develops grad-ually; although the displacement in the streamwise direction between particles inthe boundary layers and those in the main flow is cumulative with time, the flowis laminated and the SPH particles remain ordered till the end of the simulation(t = 6.79 seconds). The corresponding turbulence-related quantities for this condi-tion at time ∼5.24 seconds are displayed in Figure 10.8. The representation of themean velocity profile takes on a nearly perfect plug flow: even for the fluid particlesclose to the wall boundaries, their velocities approach the bulk velocity, 1.0 m/s.Compared to those shown in Figure 10.7, both the velocity fluctuations in the xand y direction are restrained, even when the time is extended by approximately3.3 seconds beyond that presented in Figure 10.7; for these conditions the



fluctuations are at a much lower level, namely: u′u′ is within 0 to 1.5 × 10−14 m2/s2

and v′v′ within 0 to 2.0 × 10−8 m2 s2. The velocity fluctuation in either the x ory direction gets effectively damped, and although the electromagnetic force pre-scribed in the y direction diminishes, the restraining effect exerted on the velocityfluctuations in this direction seems to become stronger, not weaker. This is seem-ingly unreasonable; however, it may be explained on the basis of what is observedin Figure 10.7: the velocity fluctuations in the x direction are not constrained by theapplied streamwise magnetic field, thus they have very large values; these velocityfluctuations in the x direction transfer “turbulent” energy to the velocity fluctua-tions in the y direction, which, in turn, also acquire large values. This supports theviewpoint advanced above: an effective MHD method for restraining the transitionto turbulence needs to suppress simultaneously the velocity fluctuations in the xand y direction.

Ten SPH simulations were performed for θ= 0, 10, 20, 30, 40, 50, 60, 70, 80,and 90, respectively. All the results are listed in Table 10.1. An applied streamwisemagnetic field (θ= 0) is more effective in restraining the velocity fluctuations inthe y direction, but it has no influence on the velocity fluctuations in the x direction.

Table 10.1. Restraining effect upon the transition to turbulence by the tilting angleθ (Re = 104, St = 40)

θ() Onset of turbulence Turbulence-related quantitiesat time equal ub u′u′ v′v′to (seconds)

c90 0.96 0.991 0–4.3 × 10−11 0–0.068c80 1.02 0.99 0–1.1 × 10−11 0–0.014c70 1.23 0.989 0–4.2 × 10−12 0–0.011c60 1.85 0.988 0–1.1 × 10−12 0–0.006a50 – 0.985 0–5.4 × 10−17 0–9.2 × 10−9

a40 – 0.978 0–5.1 × 10−18 0–1.2 × 10−8

a30 – 0.965 0–5.7 × 10−18 0–2.2 × 10−8

a20 – 0.93 0–1.5 × 10−14 0–2.0 × 10−8

a10 – 0.78 0–8.0 × 10−7 0–3.5 × 10−8

c0 1.95 0.55 0.015–0.06 0–2.1 × 10−3

a Up to the end of simulation time, 6.79 seconds, turbulence is practically not detected.The turbulence-related quantities are calculated at time ∼5.24 seconds.b In the central region of the fluid flow, the average velocity is about 1.0 m/s. The datumshown here is the velocity of the fluid particle closest to the wall.c The turbulence-related quantities are calculated shortly before the onset of turbulence.For instance, the turbulence quantities shown in the last row are calculated around at1.90 seconds.



It is the fast growth of the velocity fluctuations in the x direction that leads to theearly turbulence onset (1.95 seconds). With increasing θ (θ= 10 to 50), the meanflow velocity profile approaches the plug flow more closely; the velocity fluctuationsin the x direction are effectively restrained due to the increasing electromagneticforce produced in this direction; although the y-component of the electromagneticforce decreases, the restraining effect exerted on the velocity fluctuations in the ydirection benefits from the effective constraint of its counterparts in the x direction.Further increase of θ (θ= 60 to 80) yields the renewed appearance of turbulence,largely due to the increase of the velocity fluctuations in the y direction. An appliedtransverse magnetic field (θ= 90) is more effective in controlling the mean flowprofile, but it does not restrain well the transition to turbulence. Turbulence occursat an earlier time (0.96 seconds), when the applied magnetic field is aligned withthe transverse direction.

Enhanced conditioning effect of the transition to turbulence is obtained whenthe oblique angle θ is within the range 20 to 50. In this situation, the mean flowvelocity profile is close to a perfect plug flow; the velocity fluctuations in both the xand y direction are kept at very low levels, namely: u′u′ is below 1.5 × 10−14 m2/s2

and v′v′ less than 2.2 × 10−8 m2/s2; and, effectively, the onset of flow turbulencedoes not take place up to the end of the simulation time (6.79 seconds).

SPH simulations were also conducted with a magnetic field, B, increased by afactor of 10 to give a Stuart number, St, of 4,000 for the MHD flow of Re = 104 withthe θ varying in the range of 0 to 90. For the range 10 ≤ θ≤ 80 the transition toturbulence gets effectively suppressed; even so, for the magnetic supply conditionsof θ= 0 and 90 with this increased magnetic field applied, the flow turbulencestill appears, which supports the viewpoint advanced above; an effective MHDmethod for restraining the transition to turbulence requires a strategy that restrainsall velocity fluctuation components simultaneously.

10.4 SPH Numerical Modeling for Ballistic-DiffusiveHeat Conduction

Design of nanoscale systems, such as semiconductor devices based on the GaAsMESFETs or Si MOSFETs, and ultrafast (picoseconds or even femtoseconds)pulsed lasers have forced a fresh look into fundamental heat transfer issues [26].At nanoscale level, the classical Fourier heat diffusion is not valid due to: (a)the mean free path of the energy carriers becomes comparable to or larger thanthe characteristic length scale of the particular device/system, and/or (b) the timescale of the processes becomes comparable to or smaller than the relaxation timeof the energy carriers. Numerical solutions either using the Boltzmann transportequation (BTE) or using atomic-level simulations such as the molecular dynamicssimulation (MDS) and Monte Carlo simulation (MCS) are helpful in understand-ing the physics of heat transfer in this regime; however, these options requirelarge computational resources, which make them not very effective analyticaltools for the design/management of devices in practical nanoscale engineering.



The Cattaneo-Vernotte (C-V) hyperbolic equation [27], for a while, seemed to bea convenient alternative approach, since it excludes the so-called infinite thermalpropagation speed assumption implied in Fourier law. The work reported by Haji-Sheikh et al. [28] was the first one to point out certain anomalies existing in thehyperbolic solutions. Further on, recent work by Jiang and Sousa [29] also demon-strated the presence of hyperbolic anomalies in a hollow sphere with the Dirichletboundary condition prescribed. Although may exist some remedies [28] to the C–Vequation, a newly developed heat conduction formulation, known as the ballistic-diffusive equation [30, 31], seems to be a viable candidate for applications in thisparticular field.

The key concept of the model, on which the ballistic-diffusive heat conduc-tion equation is formulated, is to split the heat carriers inside the medium into twocomponents – ballistic and diffusive. The ballistic component is determined fromthe prescribed boundary condition and/or nanoscale heat sources, and it only expe-riences outscattering; the transport of the scattered and excited heat carriers insidethe medium is treated as a diffusive component. This approach has its origin inmethodologies used to deal with radiative heat transfer [32, 33].

The ballistic-diffusive equation is derived on the basis of the BTE under therelaxation time approximation; it differs, in appearance, from the C–V equationmainly by having the additional ballistic term, namely

τ∂2um

∂t2 + ∂um

∂t= ∇

(λ

C∇um

)− ∇ · qb +

(S + τ ∂S

∂t

)(36)

where, C refers to the volumetric specific heat (J/m3 K), qb to the ballistic heatflux vector (W/m2), S to the heat generation (W/m3), t to time (seconds), um to theinternal energy per unit volume (J/m3), λ to the heat conductivity (W/m K), and τto the thermal relaxation time (seconds). The heat generation S is assumed to havea much larger feature size than the mean free path of the heat carriers; otherwise itshould be treated as a ballistic term also. In the original work of Chen [30, 31], thefollowing general boundary condition for the diffusive components is as follows:

τ∂um

∂t+ um = 2

3∇um · n (37)

where, n (bold text designates vector) is the inward unit vector perpendicular to theboundary, and is the mean free path of heat carrier (m). The ballistic heat flux qb

is given by an algebraic analytical expression [31, 34]; however, the full solutionof equation (36) still has to resort to an appropriate numerical method, in general,the finite difference or finite element method.

The present section aims to show the viability and accuracy of the SPH solutionof the one-dimensional (1-D) model proposed by Chen [31]. It should be mentionedthat SPH was already employed to solve the Fourier heat conduction equations withsatisfying results [3, 4, 35], and a corrective SPH method [35] is used by Chen andBeraun [36] to solve the non-classic heat transfer process in thin metal films heatedby ultrashort laser pulse [37].



10.4.1 Transient heat conduction across thin films

In Chen [31] it is reported that the application of the ballistic-diffusive model to atransient heat conduction process in thin films. The same thermal case is consideredin the present study: a thin (nanoscale) film of thickness L with constant thermalproperties initially at ambient temperature T0; at the time t ≥ 0, one boundarysurface emits phonons at temperature T1; both boundary surfaces are black bodyemitters and there is no heat generation inside the film. The dimensionless ballistic-diffusive equation and associated initial and boundary restrictions can be writtenas in Chen [31].

∂2θm

∂t∗2 + ∂θm

∂t∗= Kn2

3

∂2θm

∂η2 − Kn∂q∗

b

∂η(38)

t∗ = 0, θ(η, 0) = 0∂θ(η, t∗)

∂t∗

∣∣∣∣t∗=0

= 0 (39)

η = 0,(∂θm

∂t∗+ θm

)η=0

= 2Kn

3

(∂θm

∂η

)η=0

(40)

η = 0, θb1 = 1 (41)

η = 1,(∂θm

∂t∗+ θm

)η=1

= −2Kn

3

(∂θm

∂η

)η=1

(42)

η = 1, θb2 = 0 (43)

where, η and θm are the dimensionless coordinate and diffusive temperature,respectively.

The boundary heat flux, its derivative, and the temperature of the ballistic com-ponents have their algebraic analytical expressions [31] given in non-dimensionalform as:

q∗b(η, t∗) =

⎧⎪⎨⎪⎩12

∫ 1

µt

µe−(η/µKn)dµ(0 ≤ µt = x

vt≤ 1

)0 other µt

(44)

∂q∗b

(η, t∗

)∂η

=

⎧⎪⎨⎪⎩− 1

2Kn

∫ 1

µt

e−(η/µKn)dµ+ η

Knt∗2 e−t∗ (0 ≤ µt ≤ 1)

0 other µt

(45)

θ∗b(η, t∗

) =

⎧⎪⎨⎪⎩12

∫ 1

µt

e−(η/µKn)dµ (0 ≤ µt ≤ 1)

0 other µt

(46)



where, µt = x/(vt), v refers to the carrier group velocity (m/s), x to the coordinate(m), and µ to the directional cosine.

The non-dimensional parameters are defined as follows:

θm = um − um0

CT, θb = ub − ub0

CT, q∗

m = qm − qm0

CvT, q∗

b = qb − qb0

CvT,

(47)

t∗ = t/τ, η = x/L

with

Kn = /L, T = T1 − T0 (48)

where, Kn refers to the phonon Knudsen number, L to the thickness of the nanoscalethin film (m), T0 and T1 are the initial temperature (K) and the phonon emissiontemperature at the left boundary (K), respectively.

Thus, the total temperature and heat flux are defined by:

θ = u − u0

CT= T − T0

T, q∗ = q − q0

CvT(49)

The diffusive component of the heat flux is governed by the following equation.

∂q∗m

∂t∗+ q∗

m = −Kn

3∇θm (50)

where, q∗m is the dimensionless diffusive heat flux.

In Chen [31] the above set of equations were solved numerically with the finitedifference method (FDM) and the results were compared against those obtainedusing: (a) the direct solution of the BTE, (b) the Cattaneo hyperbolic heat conduc-tion equation [27], and (c) the classic Fourier heat diffusion equation. Relative tothe BTE results, the ballistic-diffusion heat conduction equation gives consistenttemperature and heat flux profiles, whereas the Cattaneo and Fourier heat conduc-tion model both lead to erroneous representations of the processes. In the presentstudy, the ballistic-diffusive equation is to be solved with the SPH method.

10.4.2 SPH modeling

In the implementation of the SPH method to solve the heat conduction, the thermalaction at time t is represented by a collection of N particles located at a position ri,i = 1, 2, ……, N . The “smoothed” value of any field quantity f (r, t) at a space pointr and at a time t is a weighted sum of all contributions from neighboring particles

〈 f (r, t)〉 =N∑

j=1

Vj f (rj, t)w(|r − rj|, h

)(51)

where, N refers to the total SPH particle number.



Since the particle density remains constant here, the weight is the volume of SPHparticle instead of the quotient of mass and density [9]. The first order derivativeof f (r, t) is written as:

〈∇f (r, t)〉 =N∑

j=1

Vjf (rj, t)∇w(|r − rj|, h) (52)

Equation (52) is not symmetric; however, by using the following mathematicaloperator

∇f = ∇ (χ f )− f ∇χχ

(53)

where, χ is unity, equation (52) can be rewritten in the following symmetric form:

〈∇f (r, t)〉 =N∑

j=1

Vj[ f (r, t)− f (rj, t)]∇w(|r − rj|, h) (54)

In terms of equation (54), the “smoothing” formulation of the first-order derivativeof the temperature (θm) is:

∇θm(ηi, t∗) =∑

j

Vj[θm(ηi, t∗) − θm(ηj , t∗)]∇iwij (55)

where, ∇i denotes the gradient taken with respect to the coordinates at particle i

∇iwij = ηij

|ηij|∂wij

∂ηi(56)

and ηij = ηi − ηj and wij = w(|ηi − ηj|, h).One of the key issues for the SPH modeling of the ballistic-diffusive heat con-

duction is how to construct the “smoothing” formulation for the second derivative ofthe diffusive temperature (θm) in equation (36). A formulation, which combines thestandard first-order SPH derivative (equation (55)) and the finite difference concept,was first used by Brookshaw [38] and later on by Cleary and Monaghan [3]. Thisprocedure is computationally efficient, since it only involves the first-order deriva-tive of the smoothing kernel. Although it does not conserve the angular momentumaccurately, its use in simulations of Fourier heat conduction problems [3] has beenvery successful. It is employed in the present study and has the following evolutionform:

∂2θm(ηi, t∗)

∂η2 =∑

j

2Vj[θm(ηi, t∗) − θm(ηj, t∗)]1

|ηij|∂wij

∂ηi(57)

Five hundred SPH particles are uniformly distributed along the heat transferpath. The separation distance between neighboring particles is p (= L/500). The



W

0

∆p/2 ∆pη

A1

A1 = 1, A2 < 1, A2 + A3 = 1

A = ∫ w( η − η′ , h)dη′

A2

A3

Figure 10.9. Illustration of SPH “near-boundary deficiency” and the boundarytreatment devised.

inner-node approach is used, which means that either the first or the last SPHparticle is not set on the boundary surfaces, but has half p distance away fromthe respective adjoining boundary. The time derivatives are approximated by theexplicit central (for the second-order derivative) or by the forward (for the first-order derivative) scheme. The time step is adjusted to satisfy the numerical stabilityrequirements.

In this implementation, the high order 3-splines quintic kernel is also used.

10.4.3 Boundary treatment

In the present ballistic-diffusive heat conduction case, the temperature specifiedat the boundary does not represent the equivalent equilibrium temperature of theboundary; the boundary temperature and the boundary flux are calculated fromthe summation of the corresponding ballistic and diffusive components. Therefore,the temperature values of all the SPH particles depend on the smoothing interpo-lation. This is the reason why the boundary condition requires special treatment.

On the basis of the properties of the smoothing kernel, a simple boundarytreatment is devised, as illustrated in Figure 10.9.

One of the properties the SPH smoothing kernel must have is the normalizationproperty [15], that is ∫

w(|η − η′|, h)dη′ = 1.0 (58)

Clearly, for the particles close to the boundary the normalization property is notsatisfied, which is the reason for the SPH “near-boundary deficiency”. An expedientboundary treatment can be devised by varying the normalization constant nd valuesto meet the requirements set by equation (58) for all the particles. A general formfor the SPH smoothing kernel is:

w(s, h) = ndF(s, h) (59)

where, s = |ηij|/h, and F is a fitting function.



By substituting equation (59) into equation (58), the following expression,which is used to adjust the values of the normalization constant of the smoothingfunction, nd, yields:

nd = 1.0∫F(s, h)dη′ (60)

Equation (60) is essentially equivalent to the corrective interpolation formulationfor a field quantity of zero-th order proposed by Liu et al. [15], which gives for the1-D 3-spline quintic kernel, nd [39]:

nd = 1

120×

⎧⎪⎪⎨⎪⎪⎩1.4632 1st closest particle to the boundary1.0781 2nd closest particle to the boundary1.0054 3rd closest particle to the boundary1.0 other particles

(61)

The particle closest to the boundary gets the largest modification factor, 1.4632,which implies that the “near-boundary deficiency” stays primarily with the closestparticle to the boundary.

10.4.4 Results

Figures 10.10 and 10.11 display the non-dimensional diffusive temperature andheat flux distributions in thin films of different thicknesses yielding for Kn valuesof 10 and 1.0, respectively. The overall agreement between the SPH and FDMcalculations is excellent and more accurate predictions can be given by SPH for theheat conduction processes in solid films with smaller values of Kn.

The distributions for the total (summation of the ballistic and diffusive com-ponents) temperature and heat flux were also calculated; the ballistic componentdominates and due to its analytical formulation yields enhanced agreement betweenthe SPH and FDM results, as shown in Figure 10.12.

As seen from Figure 10.12, the calculated boundary temperature is not con-stant, but increases as time advances. In Chen [31] this artificial temperaturejump is attributed to the inconsistency between the temperatures obtained in theballistic-diffusive calculation and the imposed boundary temperature. As men-tioned previously, the imposed boundary temperature is the temperature of theemitted phonon, not the equivalent equilibrium temperature at the boundary; how-ever, the calculated temperature is the temperature of equilibrium phonons that canbe understood as a measure of the local internal energy. The so-called equivalentequilibrium concept can be visualized as the adiabatic thermalization equilibriumstate of the local phonons. In what concerns the SPH calculation, the temperatureof the particle closest to the boundary cannot be determined directly from the spec-ified temperature of the emitted phonons at the boundary, but depends on the SPHsmoothing interpolation. This fact explains why further treatment of the boundary



Kn = 1.0Lines : FDMSymbols : SPH t* 0.1

t* 0.01

t* 1.0

Non-dimensional position

Non

-dim

ensi

onal

diff

usiv

e te

mpe

ratu

reN

on-d

imen

sion

al d

iffus

ive

heat

flux

0.00.0

0.01

0.02

0.03

0.04

0.05−0.02

−0.01

0.00

0.01

0.02

0.2 0.4 0.6 0.8 1.0

t* 0.1

t* 0.01

t* 1.0

Figure 10.10. Comparison between the SPH and the FD predictions for non-dimensional diffusive temperature and heat flux with Kn = 10.

Kn 1.0Lines : FDMSymbols : SPH

t * 0.1

t * 0.1

t * 0.1

t * 0.1

t * 10

t * 10

Non

-dim

ensi

onal

diff

usiv

e te

mpe

ratu

reN

on-d

imen

sion

al d

iffus

ive

heat

flux


0.00.00

0.05

0.15

0.25

−0.12

−0.09

−0.06

−0.03

−0.00

0.03

0.06

0.09

0.10

0.20

0.30

0.2 0.4 0.6 0.8 1.0

Figure 10.11. Comparison between the SPH and the FD predictions for non-dimensional diffusive temperature and heat flux with Kn = 1.0.



Kn 1.0Lines : FDM

Symbols : SPHt * 1.0

t * 1.0

t * 1.0


Non

-dim

ensi

onal

tem

pera

ture

Non

-dim

ensi

onal

hea

t flu

x

0.00.0

0.2

0.4

0.6

0.8

1.00.00

0.05

0.10

0.15

0.20

0.25

0.2 0.4 0.6 0.8 1.0

t * 1.0

t * 10

t * 10

Figure 10.12. Comparison between the SPH and the FD predictions for non-dimensional total temperature and heat flux with Kn = 1.0.

is required. Figure 10.13 displays the contributions of the ballistic and diffusivecomponents to the total temperature (internal energy) and heat flux. The SPH cal-culations reproduce the characteristics of the ballistic-diffusive heat conductionreported in Chen [31].

Close observation of the calculation source data for all the 500 SPH particlesreveals an anomaly for the SPH heat flux result for the second particle away fromthe boundary. The heat flux of this particle is inconsistent with the overall trend ofthe heat flux profile and clearly deviates from the FDM predictions. This particlewas not shown in Figure 10.10 due to the limited amount of particles (only 50)selected. The implementation of equation (61) can affect the symmetry of the SPHformations, in particular that given by equation (55), which may explain the causefor this anomaly. A test simulation demonstrates a possible remedy for this anomalyis the use of a ad hoc operation: artificially increase the nd of the second particleclose to the boundary to 1.234 (mid value of the nd-s for the first and third particleaway from the boundary), which yields [39]

nd = 1120

×

⎧⎪⎪⎨⎪⎪⎩1.4632 1st closest particle to the boundary1.234 2nd closest particle to the boundary1.0054 3rd closest particle to the boundary1.0 other particles

(62)



Kn 1.0, t 1.0Solid Lines : FDM

Symbols : SPH

0.00.0

Non

-dim

ensi

onal

tem

pera

ture


Artificial wave front

Artificial wave front

Diffusivecomponent

Diffusivecomponent

Ballistic component

Ballisticcomponent

Total

TotalN

on-d

imen

sion

al h

eat f

lux

0.1

0.2

0.3

0.4

0.5

0.6−0.1

0.0

0.1

0.2

0.3

0.2 0.4 0.6 0.8 1.0

Figure 10.13. Contribution of the ballistic and diffusive components to the non-dimensional total temperature and heat flux.

Increasing the nd value of the second closest particle to the boundary improves thesymmetry contributions among the particles in the vicinity of the boundary region,in particular the symmetry between the first and second closest particles to theboundary. By plotting all the SPH particles near the left boundary, Figure 10.14clearly indicates the limitation of the treatment formulated by equation (61).

The use of equation (62) provides an effective ad hoc modification for the ndby eliminating the observed heat flux anomaly and it does not influence the overallcalculation accuracy.

10.5 Mesoscopic Pore-Scale SPH Model for Fluid Flowin Porous Media

Many industrial applications involve flow in porous media, such as oil exploration,groundwater purification from hazardous wastes, and packed bed chemical reac-tors. One application area of particular interest is in liquid composite molding



0.04Kn = 10SPH with boundary treatmentexpressed by eqation (61)

SPH with boundary treatmentexpressed by eqation (62)

FDM

t * = 1.0

t * = 0.1

t * = 0.1

t * = 1.0

t * = 0.01

t * = 0.01

0.03

0.02

0.01

0.00

0.000

−0.005

−0.010

−0.015

−0.020

0.000

Non

-dim

ensi

onal

diff

usiv

ehe

at fl

uxN

on-d

imen

sion

al d

iffus

ive

tem

pera

ture

0.010 0.020


0.030

Figure 10.14. Error resulting from the boundary treatment expressed by equa-tion (61) and the ad hoc correction effect by using equation (62).

(LCM), a process being widely used to manufacture polymer composite compo-nents in the civil, aerospace, automotive and defense industries [40]. ImportantLCM technologies are the resin transfer molding (RTM) and the structural reactioninjection molding (SRIM), and for both cases the understanding of the physics ofliquid flow in the porous preform is a requirement for the design of the mold andthe choice of the operation procedure, which are critical to the quality of the finalproduct. To this goal CFD numerical modeling can effectively reduce the experi-mental parametric investigation effort needed for the optimization of the industrialprocess. Therefore, it can play an important role in practice. CFD modeling of fluidflow in porous media is in general conducted by using Darcy’s law [41, 42], whichis expressed in the following form:

〈v〉 = −Kµ

∇P (63)



where, <v> and P are the volume averaged (superficial) velocity and the pressureof the fluid (bold text denotes vector or tensor); K is the permeability tensor of theporous medium; and µ is the dynamic viscosity of the fluid.

Darcy’s law is a macroscopic phenomenological model, which is largely basedon experimental observations. In this law, the complex interactions between fluidand microscopic porous structures are all lumped in a macroscopic physicalquantity – the permeability tensor, K. As the permeability tensor is of paramountimportance to the fluid flow in porous media, numerous research efforts [42–45]were devoted to establish theoretical relations between the permeability and othercharacteristic material properties with the purpose of avoiding time-consumingphysical experiments. These theoretical models are generally based on the analysisof the microscopic porous structure of the medium and the subsequent extractionof the desired macroscopic information. The complicated, diverse pore structuresin reality present a major challenge to pure (or semiempirical) theoretical modelsbecause they are all constructed on the basis of artificially simple, regular porearrangements.

Obviously, the modeling of fluid flow in fibrous porous media directly fromthe mesoscopic pore structure level provides refined fluid flow information andit does not resort to Darcy’s law. Moreover, the simulated fluid field is amenableto the determination of K as well. Over the last few decades, rapid advances incomputer capabilities and computational algorithms enabled this kind of modelingwork. For example, the lattice-gas-automaton (LGA) or lattice Boltzmann method(LBM) [46], based upon a micro- or mesoscopic model of kinetic formulationsor the Boltzmann equation, respectively, can bridge the gap between microscopicstructures and macroscopic phenomena. These models have the advantage of allow-ing parallelism in a straightforward manner for large-scale numerical calculations,and they have the attractive feature of nonslip bounce-back solid boundary treat-ment for the simulation of fluid flow in porous media. The simulations based onLGA [47, 48] and LBM [49–51] have demonstrated that Darcy’s law can be repro-duced by these methods. The simulated porous media in the work by Koponen etal. [49] describe three-dimensional (3-D) random fiber webs that closely resemblefibrous sheets such as paper and non-woven fabrics. The computed permeability ofthese webs presents an exponential dependence on the porosity over a large rangeof porosity and is in good agreement with experimental data [52, 53]. Spaid andPhelan [51] investigated the resin injection process encountered in RTM. The cellpermeability obtained for transverse flow through regularly arranged porous towsof circular or elliptical cross-section agrees well with the semi-analytical solution.In contrast to LBM or LGA, SPH is a meshless particle-based method and offersmore freedom in dealing with complicated geometries. Application of SPH to fluidflow in porous media has the potential of providing a mesoscopic/microscopicpore-scale insight into the relevant physics [54–57]. In the present work, SPHis employed to construct a mesoscopic pore-scale model for fluid flow in porousmedia, in particular, for the transverse fluid flow in randomly aligned fibrous porousmedia.



10.5.1 Modeling strategy

SPH formulation and methodologySPH formulation is directly based on the resolution of the macroscopic governingequations of fluid flow. Derivation of the SPH formulation from traditional Eulerian-based equations is routine, as described in Section 2; therefore, in this section isonly briefly surveyed.

In SPH, the continuous flow at time t is represented by a collection of N particleslocated at position rn(t) and moving with velocity vn(t), i = 1, 2, . . . . . . , N . The“smoothed” value of any field quantity q(r, t) at a space point r is a weighted sumof all contributions from the neighboring particles

〈q(r, t)〉 =N∑

j=1

mj

ρ(rj)q(rj , t)w(|r − rj|, h) (64)

where, mj and ρ(rj) denote the mass and density of particle j, respectively. w(|r|, h)is the weight or smoothing function with h being the smoothing length. In thepresent SPH implementation, the high-order 3-splines quintic kernel [15] is usedand the smoothing length h is equal to

√2S, withS being the average separation

of the SPH particles. In the SPH formulation the gradient of q(r, t) is determined as:

〈∇q(r, t)〉 =N∑

j=1

mj

ρj[q(rj , t) − q(r, t)]∇w(|r − rj|, h) (65)

Equation (65) is already symmetrized [24], and for the second order viscousdiffusive term, the SPH theory gives the following formulation [8, 24]:

1

ρ∇ · (µ∇q) =

∑j

mj

ρnρj(µn + µj)qnj

1

|rnj|∂wnj

∂rn(66)

where, qnj = qn − qj, rnj = rn − rj and wnj = w(|rn − rj|, h).In terms of equation (65), the “smoothed” version of the continuity equation is:

dρn

dt=

∑j

mjvnj · rnj

|rnj|∂wnj

∂rn(67)

The summation is over all neighboring particles j with exception of particle n itselfand vnj = vn − vj.

The transformation of the N–S equations into their SPH version is relativelyinvolved, because it requires the symmetrization of the pressure terms [24] to satisfymomentum conservation, and the adequate treatment of the viscous terms. In SPH,



the symmetrization version of pressure gradient can be formulated as:

1ρn

∇pn =∑

j

mj

(pn

ρ2n

+ pj

ρ2j

)rnj

|rnj|∂wnj

∂rn(68)

From equations (66) and (68), the standard SPH form of N–S equation is derived [8]

dvn

dt= −

∑j

mj

(pn

ρ2n

+ pj

ρ2j

)rnj∣∣rnj

∣∣ ∂wnj

∂rn+

∑j

mj(µn + µj

)vnj

ρnρj

(1

|rnj|∂wnj

∂rn

)+FB

(69)

where, FB denotes an external body force. In this work, the following SPHexpression for the momentum equations is used.

dvn

dt= −

∑j

mj

(pn

ρ2n

+ pj

ρ2j

+ RGηnj

)rnj

|rnj|∂wnj

∂rn(70)

+∑

j

mj(µn + µj)vnj

ρnρj

(1

|rnj|∂wnj

∂rn

)+ FB +

∑f

Equation (70) was proposed by Jiang et al. (2007) and it differs from the standardSPH N–S equation in what concerns two terms on the right-hand side of this equa-tion, namely: (a) RGηnj , an artificial pressure, which is used to restrain the tensileinstability [58]; (b)

∑f , an additional force, which is introduced with the purpose

of correctly mimicking the no-penetrating restraint prescribed by the solid porestructure. The term RGηnj is calculated as follows:

Gnj = wnj

w(S, h)(71)

R =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

ϕ1|pn|ρ2

nif pn < 0

ϕ1|pj|ρ2

j

if pj < 0

ϕ2

(|pn|ρ2

n+ |pj|ρ2

j

)if pn > 0 and pj > 0

(72)

The factors ϕ1 and ϕ2 are taken to be 0.2 and 0.05, respectively. The exponentialη is dependent on the smoothing kernel as η≥ w(0, h)/w(S, h); 3.0 is adoptedfor η in this work. The repulsive force f (per unit mass) is initiated from a porousmaterial particle and is applied to the neighboring fluid particles within a distance



less than a threshold value r0. In these conditions, for most of the fluid particles,∑f = 0. f is assumed to be of the 12-6 Lennard–Jones pair potential form, namely

f (r) = φFB

[( r0

r

)12 −( r0

r

)6]

r

r(r < r0) (73)

Determination of the constants: φ and r0 were discussed in detail in Jiang et al.[55]; the values r0 = 0.6S and φ= 15.0 were found to be suitable for the presentSPH implementation.

The quasi-compressible method, presented in Section 2 is used to comply withthe incompressibility condition for the fluid. The values of p0 and cs are taken inaccordance with the following relation

c2s = γp0

ρ0= max

(0.01U 2, 0.01FBL,

0.01Uµ

ρ0L

)(74)

where, U and L are the characteristic (or maximum) fluid velocity and thecharacteristic length of the geometry, respectively.

The positions of SPH particles are determined directly from the flow field using

drn

dt= vn (75)

A direct numerical integration algorithm is used to perform the integration of equa-tions (67), (70), and (75). The time stept is adaptive and determined by the CFLcondition [8]

t = min

⎛⎜⎝ 0.125h

maxn

(∼100.0U , |vn|) , 0.25

√√√√ h∣∣∣ dvndt

∣∣∣ , 0.25h

cs,n

⎞⎟⎠ (76)

On the right-hand side of equation (76), the quantity ∼100.0U is introduced inthe denominator of the first constraint to restrict it to a small value so that thecalculation advances mostly in a fixed small time step. In this way the accuracyof the integration procedure can be enhanced and the calculation stability is alsoimproved.

Mesoscopic pore structuresThe porous matrix is established by randomly packing a certain number of fibers.Only the transverse flow across the fibers is simulated in the present implemen-tation. Typical 2-D porous microstructures are illustrated in Figure 10.15. Thecross-sectional shape of the fibers is either circular (Figure 10.15a) or square(Figure 10.15b), and the fibers can be either overlapped with no precondition-ing (Figures 10.15a and b) or be isolated from each other (Figure 10.15c). Eachporous system is formed by packing fibers of fixed cross-section (shape and area).



(c)

(a) (b)

Figure 10.15. Typical 2-D porous microstructures formed by randomly packing anumber of fibers: (a) fibers are of circular cross-section with unlim-ited overlapping between fibers; (b) fibers are of square cross-sectionwith unlimited overlapping between fibers; (c) fibers are of circularcross-section and the fibers are isolated from each other. Void spacerepresents the fluid region.

Different porous systems have fibers of different cross-sectional shapes, differentcross-sectional areas or different relative arrangement conditions.

Both the resin fluid and the fibers are represented by SPH particles.The positionsof the SPH particles used to represent the solid pore matrix are fixed throughoutthe simulation. Thus, the porosity ε of a porous system is determined as:

ε = 1 − number of fixed particlestotal number of particles

(77)

Fixed porous material particles contribute to the density variations and exert viscousforces upon the nearby fluid particles, while their densities are kept constant. Theno-penetrating restraint prescribed by solid pore structure is ensured by activatinga repulsive force, as expressed by equation (73), on a fluid particle once it is awayfrom a porous material particle by a distance less than r0. With this additional force,the fluid particles are directed to pass by the pore structures in physically acceptablepaths [55].

Other relevant conditionsThe simulated domain is a 2-D square of dimensions 10 mm × 10 mm with periodicboundary conditions applied in both the streamwise and transverse flow directions.



Initially, 100 × 100 SPH particles are distributed uniformly within the simulateddomain (i.e. x =y =S = 0.1 mm) and are all motionless; the density for allthe SPH particles is specified to be ρ0.

The fluid flow is driven by a constant body force (FBx) imposed in the streamwiseflow direction. The density, velocity, and position of each fluid particle evolve intime by integrating equations (67), (70), and (75) respectively; while all the porousmaterial particles have a fixed position and constant density (ρ0). The given physicalproperties of the resin fluid areµ= 0.1 Pa·s andρ0 = 1,000 kg/m3.The applied bodyforce per unit of mass FBx is 10 m/s2 for most of the simulations; however, when theporosity is very high, a much smaller body force (FBx = 1.0 or 0.5 m/s2) is imposedto keep the fluid flow in the creeping flow regime.

10.5.2 Results and discussion

Flow visualizationPractical LCM procedures require sufficient wetting of all the fibers by the resin.An identified serious hurdle for LCM process is the so-called preferential flow oredge flow, which is caused by inhomogeneity of the porous perform or unavoidableclearances existing between the preform edges and mold walls. Preferential flowresults in the maldistribution of injected resin and makes part(s) of the fibrouspreform not to be saturated or fully saturated, i.e., dry spot(s). To exactly predict thepreferential flow so as to optimize the resin injection process is crucial to the qualityof the final product. Pore-scale SPH model can provide fluid flow information atmicroscopic/mesoscopic pore level and it has the potential of being very usefulin practice. Figure 10.16 presents the visualization of the fluid motion across thealigned fibers due to the applied body force (FBx). Preferential flow paths areidentified by depicting the local velocity vectors. The fluid meanders in the porousmicrostructures and its path has a tortuosity determined by the location of theembedded fibers. The mesoscopic flows involved in porous systems of differentmicrostructures do not exhibit substantially different patterns and the fluid alwaysflows through the path of least resistance among the porous structures.

Derivation of permeabilityThe permeability is an important macroscopic property for fluid flow in fibrousporous media. In terms of the simulated steady state flow field, the permeability forthe transverse flow through the fibrous porous medium can be calculated as:

k = µ

ρ0FBx〈u〉 (78)

and this formulation is based on equation (63). To facilitate the comparison betweenthe present results and existing numerical results/experimental data, the permeabil-ity k (of a square domain) must be normalized. The normalization parameter istaken to be the square of the characteristic dimension, d2, of the embedded fibers;d is defined as the hydrodynamic radius of the embedded fiber’s cross-section: for



(a) (b)

(c)

Figure 10.16. Flow visualization in randomly aligned fibrous porous systems.(Note: the length scales for the velocity vectors in the three plots arenot the same and resin fluid flows from the left to the right side.) Thebackground of the plot depicts the location of the fibers. (a) Unlim-ited overlapping between fibers of 0.6 × 0.6 mm square cross section,ε= 0.705; (b) unlimited overlapping between circular fibers withcross-section of 0.8 mm diameter, ε= 0.7061; (c) isolated circularfibers with cross-section of 0.6 mm diameter, ε= 0.4661.

fibers of circular cross-section, d is the radius; while for fibers of square cross-section, d should be the half side-length of the square. It is worth pointing out thatdue to the finite numerical discretization, the circular fiber cross-section cannotbe approximated accurately and d is calculated exactly in terms of the definitionof hydrodynamic radius, instead of simply specifying it equal to the radius of thedesired circular cross-section.



Exp. data [52, 53]Exp. data [53]SPH cal

LGA cal. [48]

LGA cal. [49]Num. results [59]

1.00.90.80.7

Porosity

Dim

ensi

onle

ss p

erm

eabi

lity

0.60.50.410−3

10−2

10−1

10−0

101

Figure 10.17. A comparison between the calculated permeability for the poroussystem formed by circular fibers of 0.6 mm diameter cross-section(with unlimited overlapping between fibers) and existing numericalresults/experimental data.

The calculated dimensionless permeability for the porous system formed bypacking circular fibers of 0.6 mm diameter cross-section with unlimited overlappingbetween fibers is depicted in Figure 10.17, as a function of porosity.

The comparison between the SPH simulation results and existing experimentaldata [52, 53] indicates the present SPH calculation technique captures the mainfeatures of the fluid flow in fibrous porous media. Computational results derivedfrom other numerical techniques [48, 59] are also consistent, at least in trend, withthe present SPH simulation predictions. It should be noted that there are at leasttwo free parameters present in this comparison. First, different porous systemswere studied, namely, the LGA results [48] were obtained for 2-D porous systemswith rectangular obstacles either randomly distributed (depicted by downward opentriangles in Figure 10.17) or placed in a regular square lattice (upward open trianglesin Figure 10.17); the physical experiments [52, 53] were performed with respect to3-D compressed fiber mats and fibrous filters; the (Eulerian-based) numerical results[59] were reported for a 3-D porous system of a face-centered-cubic (fcc) array offibers. As a second free parameter for comparison, different physical propertiesof the fluid were assumed. For example, in the present study the viscosity of thefluid is specified as, µ= 0.1 Pa · s, which may be different from that used by otherresearchers, since in many cases this parameter is not reported. A nearly lineardependence of permeability on viscosity (with fixed porosity) was reported in thework by Koponen et al. [49].



From Figure 10.17, it is clearly seen that when the porosity is smaller than 0.7,the SPH-calculated dimensionless permeability is slightly larger than the FEMresult, but is noticeably larger than that obtained by LGA technique. Besidesthe above-described free parameters between different numerical efforts, there isanother reason present. For the porous systems of unlimited overlapping fibers,the embedded fibers may be connected and adjoined to each other, and they formlarger obstacles (as shown in Figure 10.15a). This holds true especially when theporosity is low. Therefore, assuming the hydrodynamic radius as the characteris-tic dimension of the individual fibers, to a great extent, underestimates the actualcharacteristic dimension of the porous system, which leads to larger dimension-less permeability values. Moreover, for porous systems established by randomlydistributed inclusions, the randomness can affect the interior non-homogeneity ofthe system, which might cause a large variation of the permeability. Koponen et al.[48] stated that the non-homogeneity could increase the permeability as much as50% as compared to a porous system with homogeneous pore structure. Notably,the SPH-calculated permeability has large departure from the LGA prediction invery low porosity range (ε< 0.5). Non-homogeneity in the structure has significantinfluence on the effective porosity when the porosity is approaching the percolationthreshold. Slight variation of the effective porosity may lead to an order of mag-nitude change of the permeability. This additional uncertainty makes it extremelydifficult to accurately predict the permeability of a porous system with very lowporosity by using any numerical technique.

The microstructures inside the porous media may have some effect on thepermeability–porosity relation. However, as displayed by Figure 10.18, for theinvestigated 2-D isotropic porous media, none of the examined participating factors:the inclusion shape (circular or square), the inclusion size and the relative arrange-ment condition between inclusions (isolated or unlimited overlapping inclusions)brings significant effect on the relation of the dimensionless permeability versusporosity.

There are some distinct features displayed by the permeability – porosity relationshown in Figure 10.18. First, the permeability has a sharp increase, as expected,when the porosity ε approaches one (k is infinite for ε= 1). Second, the permeabilityseems to follow an exponential dependence on the porosity within the intermediateporosity range, 0.5 ≤ ε≤ 0.8. A regression fit to the calculated data yields thefollowing exponential relationship:

ln(

K

d2

)= −3.13 + 9.43ε (79)

The correlation coefficient between the simulated points and the exponentially fittedcurve is 0.99. Third, the SPH simulations indicate the percolation threshold of theinvestigated porous media is smaller than 0.4. A value of 0.33 for the percolationthreshold was used [48] for a 2-D porous system consisting of randomly distributedrectangular obstacles.



Unlimited overlapping circular inclusionsof 0.6 mm diameter

Unlimited overlapping square inclusionsof 0.6mm side-length

An exponential dependenceexpressed by eqantion (79)

Unlimited overlapping circularinclusions of 0.8 mm diameterIsolated circular inclusionsof 0.6 mm diameter

0.4

10−2

10−1

100

101D

imen

sion

less

per

mea

bilit

y

0.5 0.6 0.7 0.8 0.9 1.0

Porosity

Figure 10.18. Dependence of the dimensionless permeability on pore structure.

10.6 Concluding Remarks

In Section 2 of this chapter, the SPH benchmarking conducted for flow and heattransfer situations with analytical solutions proves the reliability and feasibility ofthis methodology.

In Section 3, SPH simulations with respect to 2-D channel flow of electricallyconducting fluid were used to demonstrate that a MHD methodology to condition-ing the transition to turbulence is more effective when the magnetic field is appliedat an angle θ, with respect to the main fluid flow direction, within the range of 0(i.e., streamwise flow direction) to 90 (transverse flow direction). In what concernsthe flow at Re = 104, an applied magnetic field that gives a Stuart number of 40can effectively stop the onset of turbulence, when it forms an angle θ between 20and 50 with respect to the main fluid flow direction. For the purpose of obtainingan effective restraining effect a sufficiently strong magnetic field is required, andthe applied direction (θ) for a magnetic field of larger Stuart number also gets itseffective range enhanced. For a turbulent flow, augmented velocity fluctuations inone direction will transfer turbulent energy to the other two directions with a conse-quent cascading effect, which may spread over the whole flow region. Therefore, itcan be reasoned that an effective MHD turbulence control strategy should considerrestraining the velocity fluctuations in all flow directions simultaneously.

In Section 4, the viability of the SPH method to solve the ballistic-diffusiveheat conduction equation is demonstrated. It is reported a simple SPH procedureto give accurate temperature and heat flux representations for the ballistic-diffusiveheat conduction processes in solid thin films. This approach is primarily based on aparticular boundary treatment, which is constructed in terms of the normalization



property of the SPH smoothing kernel. It is shown that this boundary treatment canaffect the symmetry of the SPH formulations. The SPH heat flux calculation, basedon the first-order derivative of the smoothing kernel, may lead to an erroneous heatflux value at the second particle closest to the boundary. An ad hoc modification ofthe normalization coefficients of the smoothing kernel is capable of giving correctresults for all the SPH particles.

Finally, in Section 5, in what concerns the transverse fluid flow across randomlyaligned fibrous porous media, a 2-D SPH numerical model is constructed. Thedeveloped model fully utilizes the meshless characteristic of SPH methodologyand thus offers great flexibility in the construction of complicated pore structures.The proposed modeling approach enables a mesoscopic insight into the fluid flow.Pore-scale preferential flow paths are visualized, the fluid meanders in the porousmicrostructures and, as expected, always flows along the paths of least resistance.The macroscopic features of the fluid flow are fully captured by this model. Forinstance, the dimensionless permeability (normalized by the squared characteristicdimension of the fiber cross-section) is found to exhibit an exponential dependenceon the porosity in the intermediate porosity range. In addition, it is found thatneither the relative arrangement between fibers nor the fiber cross-section (shapeor area) has significant effect on this dimensionless permeability–porosity relation.

Acknowledgment

The authors gratefully acknowledge the financial support received by thisproject from FCT (Foundation for Science and Technology, Portugal) – theresearch grant POCTI/EME/59728/2004 (ACMS) and the Post-doctoral Fellow-ship, SFRH/BPD/20273/2004 (FJ), and from NSERC (Natural Sciences andEngineering Research Council of Canada) – Discovery Grant 12875 (ACMS)

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11 Evaluation of continuous and discretephase models for simulatingsubmicrometer aerosol transportand deposition

Philip Worth Longest1 and Jinxiang Xi21Virginia Commonwealth University, Richmond, VA, USA2University of Arkansas, Little Rock, AR, USA

Abstract

Submicrometer aerosols are widely prevalent throughout the environment. Exam-ples of these aerosols include combustion byproducts, ions from radioactive decay,and respiratory-specific viruses. Predicting the transport and fate of these aerosolsis important in a number of applications including large-scale environmental assess-ments, determining pollutant exposure levels in microenvironments, and evaluatingthe lung deposition and potential health effects of inhaled particles. Numerical simu-lations of submicrometer aerosols are challenging due to low deposition efficiencies,the action of concurrent inertial and diffusive transport mechanisms, and stiff equa-tion sets. As with other two-phase flow systems, both continuous and discrete phasemodels have been used to simulate the transport and deposition of submicrometeraerosols. However, each of these approaches has inherent strengths and weaknesses.In this review, the performance of continuous and discrete phase models is assessedin a series of case studies that focus on submicrometer aerosol transport and deposi-tion in the respiratory tract. The numerical methods considered include a chemicalspecies approach that neglects particle inertia, Lagrangian particle tracking, and arecently proposed drift flux model that utilizes a near-wall analytic solution. Theperformance of these models is evaluated through comparisons with experimentaldata sets of deposition in a representative double bifurcation geometry, a realis-tic model of the upper tracheobronchial region, and a realistic model of the nasalcavity. Comparisons of the mass transport methods considered with experimentaldata highlight advantages and disadvantages of each approach. These results areintended to provide guidance in selecting appropriate two-phase models for simu-lating submicrometer aerosol transport and deposition in a variety of applicationsincluding exposure analysis, respiratory dosimetry, and respiratory drug delivery.

Keywords: Modeling nanoaerosols, Submicrometer aerosol dynamics, Drift fluxmodel, Lagrangian particle tracking



11.1 Introduction

Fine respiratory aerosols can be characterized as having a particle diameter in therange of approximately 100 nm to 1 µm. Inhalable fine aerosols include dieselexhaust products, environmental and mainstream tobacco smoke, and airbornebioaerosols such as viruses and bacteria. Morawska et al. [1] reported average countmedian diameters of sample diesel exhaust and environmental tobacco smoke tobe 125 and 183 nm, respectively. The particle fraction of diesel exhaust can rangefrom 5 to 500 nm [2]. Bernstein [3] summarized a number of studies and reporteda consistent value of cigarette smoke count median diameter of approximately260 nm surrounded by a wide distribution of values ranging from 18 nm to 1.6 µm.Respiratory-specific viruses such as Avian flu and SARS typically range from 20 to200 nm [4].

Fine aerosols deposit in the respiratory tract at a reduced rate in comparison withsmaller ultrafine (<100 nm) and larger coarse (>1 µm) aerosols. This reduction indeposition for fine aerosols is due to a minimum in the net sum of diffusion, sed-imentation, and impaction effects [5]. In vivo experiments of whole-lung aerosoldeposition in humans indicate a minimum in retention for fine aerosols with a diam-eter of approximately 400 nm [6–8]. For environmental tobacco smoke and dieselexhaust, the in vivo deposition study of Morawska et al. [1] reported depositionfractions of 36 and 30%, respectively. While deposition rates are typically low forfine respiratory aerosols, environmental pollutants within this size range are oftencarcinogenic [9] and inflammatory [10]. As a result, understanding the transportand deposition characteristics of fine respiratory aerosols is critical in order tomake accurate toxicology risk assessments, to establish appropriate environmentalstandards, and to determine safe exposure limits.

Localized accumulations of particles have been widely observed at branch-ing sites in bifurcating respiratory geometries. These focal depositions, or hotspots, are often most pronounced for coarse aerosols due to inertial drivenimpaction [11–17]. However, significant increases in local deposition havealso been observed for dilute submicrometer respiratory aerosols and vapors[17,18–21]. The quantity of particles depositing within a specific localized areawill affect the transport of particles and chemical species (CS) across the mucusbarrier and ultimately the cellular-level dose received [18]. As a result, evaluationsof localized deposition values are needed to better predict cellular-level dose andresponse [22].

One advantage of computational fluid dynamics (CFD) predictions comparedwith in vivo and in vitro studies is that modeling results can readily determinelocalized deposition characteristics in three dimensions. However, due to numeri-cal challenges only few computational studies have quantified the local depositionof fine respiratory aerosols. Zhang et al. [17] compared regional and local deposi-tions of coarse and ultrafine particles, but fine aerosols were not considered. Longestand Oldham [23] modeled the effect of an upstream aerosol delivery system on thedeposition of 1 µm aerosols in a double bifurcation geometry. Robinson et al.[24] considered the total deposition of fine respiratory aerosols in a model of


Evaluation of continuous and discrete phase models 427

respiratory generations G3–G8. Longest and Oldham [25] were the first to compareCFD-predicted local depositions of fine aerosols with in vitro data.

Numerical models for simulating particle transport and deposition typi-cally employ either a Eulerian–Lagrangian [18,24,26] or a Eulerian–Eulerian[17,21,27,28] approach for submicrometer aerosol transport and deposition. TheEulerian model that is typically used for respiratory aerosols treats the particlephase as a dilute CS in a multicomponent mixture. This approach neglects particleinertia and only considers deposition due to diffusional effects. However, Longestand Xi [20] illustrated a significant influence of inertia on the regional and localdeposition of fine and ultrafine aerosols. In contrast to the CS Eulerian model,the Lagrangian approach tracks individual particles within the flow field and canaccount for a variety of forces on the particle including inertia, diffusion, gravityeffects, and near-wall interactions [29]. Disadvantages of the Lagrangian modelinclude an excessively large number of particles required to establish convergentlocal deposition profiles and a stiff equation set for fine and ultrafine particles [30].In the study of Longest and Oldham [23], a Lagrangian model was used to match theoverall in vitro deposition rate of 1 µm particles in a double bifurcation geometry;however, the model was not able to match the local deposition characteristics.

In contrast to the CS Eulerian model that is typically applied to ultrafine and finerespiratory aerosols, other continuous field two-phase flow methods are available[31,32]. Respiratory aerosols are typically dilute, such that constant flow field prop-erties can be assumed. However, finite particle inertia has been shown to stronglyaffect the deposition of fine and ultrafine particles [20]. Of the available two-phasemethods, the drift flux (DF) model is an effective approach that can be appliedto dilute systems with low Stokes numbers Stk and can account for finite particleforces [33]. Wang and Lai [34] applied a DF model to the deposition of ultrafinethrough coarse respiratory aerosols to account for gravity and electrostatic effects.However, no previous studies have considered a DF model to approximate the iner-tia of respiratory aerosols. A potential problem with the DF model applied to fineaerosols is the estimation of particle inertia in the near-wall region.

This chapter will discuss (i) the effect of inertia on the deposition of submi-crometer particles and the appropriate bounds for the application of a CS Eulerianmodel; (ii) development of a new drift flux model that effectively takes into accountfinite particle inertia; and (iii) evaluation of this new drift flux velocity correction(DF-VC) model by comparisons with experimental data and other model predic-tions in human respiratory airways. Direct comparisons between numerical andexperimental deposition results are made on a regional and highly localized basis.Computational models evaluated include a standard CS mass fraction approxima-tion, Lagrangian particle tracking, and the DF approach to account for finite particleinertia. The airway geometries considered in this study include a idealized bifurca-tion model, a cast-based tracheobronchial (TB) geometry, and an in vivo scan-basednasal cavity model. Agreement between computational and experimental resultswill help to establish an effective approach for simulating fine respiratory aerosoldeposition where both inertial and diffusional effects may be significant transportmechanisms.



11.2 Models of Airflow and Submicrometer Particle Transport

Flow fields in the respiratory tract are typically assumed to be isothermal and incom-pressible. Depending on activity conditions, the flow regime under normal breathingcan be laminar, transitional, or fully turbulent. Turbulence models have differentlevels of complexity and include direct numerical simulation (DNS), large eddysimulation (LES), the Reynolds stress model (RSM), and the Reynolds-averagedNavier–Stokes (RANS) two-equation approach. Selection of an appropriate modelfor the flow of interest mainly depends on the desired accuracy and the availablecomputational resources. DNS is the most accurate approach and resolves turbu-lent eddies at all scales, but it also requires the most computational resources.Large eddy simulations resolve large-scale energy-containing eddies, and the RSMapproach captures the anisotropic turbulence, which is significant near the wall.Even though the lower-order RANS two-equation models cannot account for tur-bulence anisotropy, these models are still often shown to adequately capture themain features of the flow, provided a very fine mesh is employed in the near-wallboundary region [35]. In this study, the low Reynolds number (LRN) k −ω two-equation model was selected based on its ability to accurately predict pressuredrop, velocity profiles, and shear stress for transitional and turbulent flows [36,37].This model was demonstrated to accurately predict particle deposition profiles fortransitional and turbulent flows in models of the oral airway [38,39], nasal pas-sages [40], and multiple bifurcations [41,42]. Moreover, the LRN k − ω modelwas shown to provide an accurate solution for laminar flow as the turbulent vis-cosity approaches zero [37]. Transport equations governing the turbulent kineticenergy (k) and the specific dissipation rate (ω) are provided by Wilcox [37] andwere previously reported in Longest and Xi [30].

11.2.1 Chemical species model for particle transport

There is an ever-increasing literature base on particle dynamics in turbulent flows,as reviewed by Crowe et al. [43]. Depending on the need to simulate particle inertiaand coupling with the continuous phase, Eulerian-based particle transport modelscan be categorized as CS, DF, or mixture models. In the respiratory system, dilutesuspensions of nanoparticles are often treated as a CS with a Eulerian mass transportmodel. This model often neglects particle inertia and the effects of the particle phaseon the flow field, i.e., one-way coupled particle transport. The transport relationgoverning the convective–diffusive motion of ultrafine aerosols in the absence ofparticle inertial effects can be written on a mass fraction basis as:

∂c

∂t+ ∂(ujc)

∂xj= ∂

∂xj

[(D + νT

ScT

)∂c

∂xj

](1)

In the above equation, c represents the mass fraction of nanoparticles, D is themolecular or Brownian diffusion coefficient, and ScT the turbulent Schmidt number



taken to be 0.9. Assuming dilute concentrations of spherical particles, the Stokes–Einstein equation was used to determine the diffusion coefficients for various sizedparticles and can be expressed as:

D = kBTCc

3πµdp(2)

where kB = 1.38 × 10−16 cm2g/s is the Boltzmann constant expressed in cgs units.The Cunningham correction factor has been computed using the expression ofAllenand Raabe [44]

Cc = 1 + λ

dp

(2.34 + 1.05 exp

(−0.39

dp

λ

))(3)

where λ is the mean free path of air, assumed to be 65 nm. The above expression isreported to be valid for all particle sizes [45]. To approximate particle depositionon the wall, the boundary condition for the Eulerian transport model is assumed tobe cwall = 0.

11.2.2 Discrete phase model

One-way coupled trajectories of monodisperse ultrafine particles ranging in diame-ter (dp) from 1 to 1,000 nm are often calculated on a Lagrangian basis by integratingan appropriate form of the particle trajectory equation. Aerosols in this size rangehave very low Stokes numbers (Stk = ρpd2

p CcU/18µD<< 1), where U is the meanfluid velocity and D a characteristic diameter of the system. Other characteristics ofthe aerosols considered include a particle density ρp = 1.00 g/cm3, a density ratioα= ρ/ρp ≈ 10−3, and a particle Reynolds number Rep = ρ|u − v|dp/µ<< 1. Theappropriate equations for spherical particle motion under these conditions can beexpressed as:

dvi

dt= αDui

Dt+ f

τpCc(ui − vi) + fi, Brownian and

dxi

dt= vi(t) (4a and 4b)

In the above equations, vi and ui are the components of the particle and local fluidvelocity, respectively. The characteristic time required for particles to respond tochanges in the flow field, or the momentum response time, is τp = Ccρp d2

p/18µ. Thepressure gradient or acceleration term is often neglected for aerosols due to smallvalues of the density ratio. However, it has been retrained here to emphasize thesignificance of fluid element acceleration in biofluid flows [29]. The drag factor f ,which represents the ratio of the drag coefficient CD to Stokes drag, is assumed tobe one for submicrometer aerosols. The effect of Brownian motion on the particletrajectories is included as a separate force per unit mass term at each time-step. Theamplitude of the Brownian force is of the form [46]

fi, Brownian = ςi

√πSo

t(5)



where ςi is a zero mean variant from a Gaussian probability density function, So aspectral intensity function directly related to the diffusion coefficient, and t thetime-step for particle integration. The influence of anisotropic fluctuations in thenear-wall region is considered by implementing an anisotropic turbulence modelproposed by Matida et al. [47], which is described as:

u′n = fvξ

√2k/3 and fv = 1 − exp (−0.002y+) (6)

In this equation, ξ is a random number generated from a Gaussian probabilitydensity function and fv a damping function component normal to the wall forvalues of y+ less than approximately 40.

11.2.3 Deposition factors

For Lagrangian tracking, the deposition efficiency (DE) is defined as the ratio ofparticles depositing within a region to the particles entering that region. For theEulerian model, the mass deposition rate on a wall can be expressed as

mw, i =N∑

j=1

−ρmAj

(D + νT

ScT

)∂c

∂n

∣∣∣∣w, j

(7)

where the summation is performed over region of interest i, and n is the wall-normalcoordinate pointing out of the geometry. Localized deposition can be presented interms of a deposition enhancement factor (DEF), which quantifies local depositionas a fraction of the total or regional DE. A DEF, similar to the enhancement factorsuggested by Balashazy et al. [11], for local region j can be defined as

DEFj = DEj/Aj∑Nj=1 DEj

/∑Nj=1 Aj

(8)

where the summation is performed over the region of interest, i.e., the upper airwaygeometries. For the Lagrangian model, the local area Aj is assumed to be a regionwith a diameter of 500 µm, or approximately 50 lung epithelial cells in length [48].The definition of the DEF is for a pre-specified constant area at each samplinglocation. Sampling locations are taken to be nodal points. Constant areas are thenassessed around each nodal point and allowed to overlap if necessary.

11.2.4 Numerical methods

To solve the governing mass and momentum conservation equations in each of thecases considered, the CFD package Fluent 6 was employed. User-supplied Fortranand C programs were implemented for the calculation of initial particle profiles,wall mass flow rates, Brownian force [30], anisotropic turbulence effect [47,49],



and near-wall velocity interpolation [30].All transport equations were discretized tobe at least second order accurate in space.A segregated implicit solver was employedto evaluate the resulting linear system of equations. This solver uses the Gauss–Seidel method in conjunction with an algebraic multigrid approach to solve thelinearized equations. The SIMPLEC algorithm was employed to evaluate pressure–velocity coupling. Convergence of the flow field solution was assumed when thenormalized global mass residuals fell below 10−5 and the residual–iteration curvesfor all flow parameters become asymptotic.

11.3 Evaluation of Inertial Effects on Submicrometer Aerosols

A direct comparison between the Eulerian and Lagrangian transport model resultscan be visualized in a branching respiratory geometry by snapshots of particlelocations at two selected times during a washout phase with a tracheal flow rateof 30 L/min (Figure 11.1). The initial velocity profile was parabolic resulting in anear-wall region of low flow beginning at the inlet. For each model, 5 nm parti-cles were initially released from an inlet plane for 0.003 seconds. The simulationwas then continued over time to illustrate the washout of the remaining particlefraction. For the Eulerian simulation at t = 0.005 seconds, a high concentrationregion is observed to interact with the first carina and inner walls of the daughterbranches (Figure 11.1a). However, these elevated concentrations have been elim-inated from the first carina by time t = 0.015 seconds due to the high velocitiesin this region (Figure 11.1b). Moderately elevated concentrations of mass fractionare observed near the second carinas at t = 0.015 seconds (Figure 11.1b). As thesimulation continues from t = 0.005 to 0.015 seconds, washout of the mass frac-tion is observed in regions of high flow, whereas regions of low flow are dominatedby mixing.

Snapshots of Lagrangian particles appear similar to the Eulerian mass transportmodel over time (Figure 11.1c and d). For time 0.005 seconds, a region of highconcentration is again observed to interact with the first carinal ridge. However,the Lagrangian particles simulated have not progressed as far as the minimumcontour level considered for the Eulerian flow (Figure 11.1a vs. c). This may be dueto the discrete nature of the Lagrangian particles and the relatively small numberof particles considered for this illustration. Approximately 2,000 particles weretracked for visualization. However, the Lagrangian model also directly accountsfor the molecular slip correction and Taylor diffusion. At time 0.015 seconds, theEulerian and Lagrangian models again appear similar (Figure 11.1b vs. d). It is notedthat the Eulerian model is presented for a mid-plane slice, whereas the Lagrangianmodel must be presented in terms of particles in the 3-D field. Therefore, the mid-plane view of the Eulerian model reveals only a thin near-wall region of elevatedconcentration in G3 at time 0.015 seconds (Figure 11.1b). In contrast, this thinnear-wall concentration layer of slow-moving particles is observed to occupy amajority of the third generation based on the 3-D view through the geometry forthe Lagrangian result (Figure 11.1d).



(a) Eulerian model at time 0.005 s

Mass fraction

0.040.030.020.010.005

0.040.030.020.010.005

(c) Lagrangian model at time 0.005 s

Mass fraction

(d) Lagrangian model at to 0.015 s

(b) Eulerian model at time 0.015 s

Figure 11.1. Comparison of Eulerian species mass fraction (a and b) andLagrangian particle distributions (c and d) at various times for aninhalation flow rate of 30 L/min in the upper airway model. A con-stant concentration of 5 nm particles was released at the inlet planefor the first 0.003 seconds of the simulation.

Figure 11.2 shows the regional deposition fractions based on both Eulerian andLagrangian particle tracking models in comparison to experimental and analyticalresults within an upper airway double bifurcation geometry (generation G3–G5)for a tracheal flow rate of 30 L/min. Predictions of the Eulerian transport modelare consistent with the correlation of Cohen and Asgharian [50] as well as the ana-lytic solution of Ingham [27] (Figure 11.2). Based on available empirical data,the correlation of Cohen and Asgharian [50] is valid for particles from 40 to200 nm and includes a high degree of variability. Over this range, good agree-ment is observed between the Eulerian and empirical models (Figure 11.2). Theparticle size and Stokes number for which the Lagrangian particle deposition frac-tion exceeds Eulerian particle deposition by a difference of 20% has been marked(Figure 11.2). This particle size is considered to be the threshold where impactionplays a significant role in total deposition fraction and will be referred to as theinertial particle diameter. At 30 L/min, the inertial particle diameter is 120 nm and



20% variation in Eulerian andLagragian simulations (120 nm)St3 = 5.0E-5

Particle diameter (nm)

Cohen and Asgharian (1990)Ingham (1991)Martonen (1993)Eulerian simulationLagrangian particleLagrangian fluid element

Dep

ositi

on fr

actio

n (%

)101

100

10−1

10−2

10−3

101 102 103

Figure 11.2. Branch-averaged deposition fraction in respiratory generationsG3–G5 of the upper airway model for an inhalation flow rate of30 L/min. Good agreement is observed between the simulated resultsand the empirical correlation of Cohen and Asgharian (1990) from 40through 200 nm. Separation of Eulerian and Lagrangian simulationresults indicates the onset of particle impaction. Particle impaction isobserved to become significant for 120 nm particles.

the associated Stokes number for generation G3 is 5.0 × 10−5. As the flow rate isincreased, the minimum particle size where impaction becomes important growssmaller due to increasing flow inertia. The smallest observed inertial particle diam-eter was 70 nm (St = 5.1× 10−5), which occurred for the highest tracheal flow rateconsidered, which was 60 L/min.

To ensure accuracy in the particle tracking routine, the deposition fraction ofmassless Lagrangian fluid elements has also been assessed (Figure 11.2). Fluidelements are assumed to have the density of the continuous fluid, or air, resulting innegligible inertia effects. The diffusion of fluid elements is based on the geometricelement diameter and is not influenced by the density. The resulting depositionfraction of Lagrangian fluid elements is observed to be in close agreement with theEulerian model results for all particle sizes considered (Figure 11.2). Therefore,differences in the Lagrangian and Eulerian solutions may be assumed to representthe effects of finite particle inertia and not numerical errors in the particle trackingroutine.

In comparison to regional-averaged values, the effects of particle inertiaon localized deposition characteristics were found to be much more dramatic[20,40,42]. As shown by Longest and Xi [20], for the upper airway bifurcation



model, inclusion of particle inertia was found to increase the maximum local micro-dosimetry factor by one order of magnitude for 40 nm particles at an inhalation flowrate of 30 L/min. The maximum DEF values predicted by the Lagrangian modelwas 103.9 in contrast to a DEFmax of 5.7 predicted by the Eulerian CS model [20]. Itwas also observed that the Lagrangian model did not appear to resolve continuouscontours of DEF below values of approximately 5.

From the above observations, it is indicated that particle inertia may be moresignificant in the regional and local deposition of fine and ultrafine aerosols thanpreviously considered. Therefore, CS Eulerian models of particle transport maysignificantly underestimate branch-averaged and local deposition values of fineand ultrafine particle deposition. This underestimation was found to be especiallysignificant for localized deposition patterns and microdosimetry estimates. For40 nm particles, a one order of magnitude increase of the DEF values was observedin Lagrangian estimates in comparison to Eulerian predictions for the upper airwaymodel at an inhalation flow rate of 30 L/min. Furthermore, the particle inertiacannot be ignored for regional deposition calculations with Stokes numbers largerthan 5.0 × 10−5 in bifurcating airways. As a result, it is necessary to develop anappropriate particle transport model that can effectively resolve continuous contoursof deposition and capture the influence of finite particle inertia for submicrometeraerosols in the respiratory tract.

11.4 An Effective Eulerian-Based Model for SimulatingSubmicrometer Aerosols

The basis of this Eulerian approach is the DF model with a novel extension to bettercompute wall deposition on a continuous basis. In contrast with the CS model, theDF approach includes particle inertial effects, which are taken into considerationthrough the convection term [25,33]

∂c

∂t+ ∂(vjc)

∂xj= ∂

∂xj

((D + νT

ScT

)∂c

∂xj

)(9)

In this equation, vj is the particle velocity, which is evaluated from the particle slipvelocity (vsj) as:

vj = vsj + uj (10)

For a continuous field solution, the particle slip velocity can be determined as afunction of inertial and gravity forces as [33]

vsj = Cc d2p

18µc(ρp − ρm)

[gj − ∂uj

∂t− ui

∂uj

∂xi

](11a)

where Cc is the Cunningham correction factor, ρp and ρm are the particle andmixture densities, dp the particle diameter, and µc the continuous field viscosity.



s

Control volumecentre (CV )

vp,iucv,nn

Particlelocation (p)

Wall boundary

Variables

n – wall normal coordinate

u – fluid velocityv – particle velocity

p – particle position

s – distance detween CV center and wall

Subscripts

CV – control volume center

Figure 11.3. Velocity components and notation within a near-wall control volume.

In equation (11a), the first term in brackets is the gravity vector and the next twoterms represent the material derivative, which accounts for fluid element accelera-tion. For the small particles considered in this study (≤1 µm), gravity effects areneglected and Stokes flow conditions can be assumed. As a result, equation (11a)can be written in terms of the fluid pressure gradient as [33]:

vsj = Cc d2p

18µc

(ρm − ρp

)ρm

∂p

∂xj(11b)

As with the CS model, the DF approach approximates perfect absorption at the wall,i.e., cwall = 0. The associated local mass deposition as a result of both diffusionaland inertial effects is expressed as:

mw,l = −ρmAlD∂c

∂n

∣∣∣∣wall

+ ρm Al c vn|wall (12a)

In the above expression, vn represents the wall-normal particle velocity scalar

vn = vi ni (12b)

where ni is the local wall-normal unit vector pointing out the geometry.Differences among the two DF approaches considered in this study are based on

how the particle velocity at the wall is computed and implemented in the particledeposition expression, equation (12a). Velocity values near the wall on a controlvolume grid and related nomenclature are indicated in Figure 11.3. For the standardDF model, slip velocities are only available at control volume center locations. Asa result, the particle velocity at the wall is approximated as the value at the centerof the nearest control volume, i.e.,

vn|wall = vcv,i ni = vcv,n (13)

As indicated in Figure 11.3, the subscript CV indicates the value at the control vol-ume center and the subscript n indicates the wall-normal scalar. Based on expectedparticle deceleration between the control volume center location and the wall



surface, the standard DF approximation is expected to significantly overestimateparticle inertia at the wall and may be mesh dependent.

To improve performance of the standard DF approach, velocity correctionsare proposed for particle conditions at the initial point of particle-to-wall contact.Instead of utilizing velocity at the near-wall cell center as with the standard DFmodel, the particle velocity at the wall is determined based on a subgrid solution ofparticle motion between the control volume center and wall location.The motivationbehind this subgrid solution is that fully resolving near-wall finite particle inertiawith a continuous model would require an excessive number of control volumes.Instead, the DF-VC model employs an analytic solution of particle velocity betweenthe near-wall control volume center location and the wall (Figure 11.3). It is assumedthat the continuous field model approximates particle diffusion within this region.Furthermore, for low particle Reynolds numbers the individual Lagrangian trans-port terms become linear and separable. Particle inertia between the control volumecenter and wall surface can then be approximated on a discrete Lagrangian basis as:

dvp, n

dt= 1

τp(up, n − vp, n) (14)

To formulate an analytic solution of equation (14), the wall-normal fluid velocityis assumed to vary linearly between the control volume center and zero at the wall.The resulting equation for particle position between the near-wall control volumecenter and the wall as a function of time can be written as

d2xp

dt2 + 1

τp

dxp

dt+ ucv,n

sτpxp = 1

τpucv, n (15)

where ucv,n is the fluid velocity at the control volume center location normal tothe wall. An analytic solution to equation (15) is possible resulting in wall-normalexpressions for particle position

xp(t) = vcv,n + λ2s

λ1 − λ2eλ1t −

(vcv,n + λ2s

λ1 − λ2+ s

)eλ2t + s (16a)

and velocity

vp,n(t) = vcv,n + λ2s

λ1 − λ2λ1eλ1t −

(vcv,n + λ2s

λ1 − λ2+ s

)λ2eλ2t (16b)

where

λ1 = 1

2

⎛⎝− 1

τp+

√(1

τp

)2

− 4(

ucv,n

sτp

)⎞⎠ (16c)

λ2 = 1

2

⎛⎝− 1

τp−

√(1τp

)2

− 4(

ucv,n

sτp

)⎞⎠ (16d)



The time for initial wall contact can be determined from equation (16a). The asso-ciated particle velocity at the point of deposition is then calculated using equation(16b) and applied to calculate the local mass deposition in equation (12a).

The expressions for near-wall particle position and velocity presented as equa-tions (16a and b) require the λ coefficients (equations 16c and d), which are rootsof the characteristic equation for particle motion, to be real and unique. If thesecoefficients are complex numbers or repeated roots, then the general solution ofnear-wall particle motion will have a different form. The occurrence of real andunique values of λ1 and λ2 requires that(

1

τp

)2

> 4(

ucv,n

sτp

)(17)

The particle response time (τp) is proportional to the particle diameter squared.Therefore, the condition for real and unique values of λ1 and λ2 is more likelyto be satisfied for smaller particles. As the particle size increases, the left-handside (LHS) of the condition decreases rapidly, which increases the probability forcomplex values of λ1 and λ2. Considering submicrometer aerosols, the limitingmaximum particle size is 1 µm, which should result in only real and unique valuesof λ1 and λ2, as verified numerically [42].

11.5 Evaluation of the DF-VC Model in an IdealizedAirway Geometry

In this section, three continuous field models were evaluated based on their abilityto capture the inertial effects of submicrometer aerosols by direct comparisonswith in vitro experimental data in a idealized bifurcation geometry. The first modelconsidered was the Eulerian CS approximation, which is known to neglect particleinertia [20]. Implementation of this model is intended as a base case to capturepurely diffusional effects. The remaining two models were based on a DF approachthat accounts for both diffusion and particle inertia effects. Differences in the DFmodels arise as a result of how the near-wall inertia is approximated, as describedin the previous section. The idealized airway bifurcation model was selected hereas a test case because it can be mathematically described, which facilitates thegeneration of identical experimental and computational geometries.

The experimental method for the generation and delivery of submicrometeraerosols was previously described by Oldham et al. [15]. Briefly, a Lovelace-typecompressed air nebulizer (In-Tox Products,Albuquerque, NM) was used to generatethe fluorescent 400 nm and 1 µm aerosols. As shown in Figure 11.4a, a customcopper enclosure [51] was used to connect the nebulizer with the double bifurcationmodel. Aerosols were dried and diluted using 5% relative humidity air at an injectrate of 10 times the nebulizer output. The aerosol was discharged to Boltzmannequilibrium by passing through a 85Kr discharger. Aerosols were pulled throughthe double bifurcation models by use of a vacuum pump. Particles that did not



(a) Aerosol delivery system (b) PRB geometry

Mixingchamber

Suddencontraction

zz

y

Outlet extension

yx

x

Inlet toPRB

Figure 11.4. Geometries considered including (a) experimental particle deliverysystem, and (b) double bifurcation model of respiratory generationsG3–G5.

deposit in the models were collected on a 25 mm diameter polycarbonate filter witha 0.1 µm pore size (Nucleopore Corporation, Pleasanton, CA). Local and totaldepositions of aerosols were determined by microscope-based counting on a gridof 1.4×0.95 mm for 1 µm particles and 1×0.69 mm for 400 nm particles [25,15].Fluorescent particle counts in photographic microscope fields were determinedusing fluorescence microscopy.

Comparisons of experimental and numerical total particle deposition valuesin the bifurcation geometry for the two particle sizes considered are illustrated inFigure 11.5. For 400 nm particles, the experimental total deposition rate is 0.0054%.The CS model appears to closely match the experimental deposition value with atotal deposition prediction of 0.0048%, resulting in a percent difference of 11.1%.In contrast, the standard DF model predicts a deposition rate that is nearly twoorders of magnitude higher than the experimental findings. As a result, the standardDF model appears to significantly overpredict the effects of particle inertia. TheDF-VC approximation matches the experimental results to a high degree with a totaldeposition value of 0.00585% and a percent difference 8.3%. As a result, both theCS and DF-VC models appear to provide adequate predictions of total depositionfor 400 nm particles. Comparisons of local deposition results with experimentalfindings will be used to evaluate differences between the CS and DF-VC models.

Experimental deposition results for 1 µm particles indicate a total depositionfraction of 0.01% (Figure 11.5), as reported in Oldham et al. [15]. Due to a lackof particle inertia in the computational model, the CS approach under predicts thetotal deposition fraction for 1 µm aerosols by one order of magnitude. In contrast,the standard DF model overpredicts the total deposition by one order of magnitude.As with 400 nm particles, predictions of the DF-VC model for 1 µm particles arein close agreement with experimental results. The DF-VC model predicts a totaldeposition value of 0.0104%, resulting in a difference of 4.0% in comparison withthe experiment.



DFDF

Exp

Tota

l dep

ositi

on fr

actio

n (%

)

CS

400 nm 1 µm

DF-VCDF-VC

Exp

CS

100

10−1

10−2

10−3

10−4

Figure 11.5. Comparison of total deposition fraction as a percentage betweenexperimental results and model predictions for 400 nm and 1 µmparticles.

Experimental results of local 400 nm particle depositions on a 1 × 0.69 mmgrid are shown in Figure 11.6a as a percentage of total deposited particles [25]. Asexpected, localized increases in particle deposition are observed at the bifurcations.Deposition contours are in the range of 1 to 5% at the first carinal ridge and 0.1 to1% at the second. A discontinuous shading pattern is observed in the first daughterbranches with contours in the range of 0.01 to 1%. At the second bifurcation,higher contour values appear to extend down the inner branch in comparison withthe outer branch. Some asymmetry in the deposition pattern may be due to minorvariations in the inlet particle concentration and alignment of the grid field with theexperimental model.

Numerical results of local particle deposition for 400 nm aerosols are reportedon a two-dimensional grid with element sizes equivalent to the experimental study,i.e., 1×0.69 mm (Figure 11.6b–d). For the CS model and 400 nm particles, hot spotsof local particle deposition are not observed at the carinal ridges (Figure 11.6b).As a result, the CS model appears to match the experimental total deposition valuebut displays significant differences from local experimental findings. Moreover,the CS model appears to predict more evenly distributed contours of depositionthroughout the physiologically realistic bifurcation (PRB) in comparison with theexperimental results. Considering the standard DF model, a significant increase inlocal particle deposition is observed at the first carinal ridge and a lesser hot spotoccurs at the second bifurcation point (Figure 11.6c). However, these hot spotsare at a higher contour level than observed in the experiment and surrounded byfew deposited particles. The DF-VC model provides the best match to localizedexperimental results for the deposition of 400 nm particles (Figure 11.6d). Max-imum localized particle deposition values at both carinal ridges are predicted tobe within the same range as observed experimentally. Specially, the DF-VC model



(a) Experiment (b) CS model and 400 nm particles

(d) DF-VC model and 400 nm particles

Percent of deposited400 nm particles 0.01 0.1 1 5

(c) DF model and 400 nm particles

Figure 11.6. Comparison between experimental and numerically determined con-tours of local deposition for 400 nm particles expressed as a per-centage of total deposition: (a) experimental, (b) CS, (c) DF, and(d) DF-VC models.

predicts localized deposition at the first and second carinal ridges to be 1 to 5% and0.1 to 1%, respectively. Furthermore, the DF-VC model predicts elevated contourswithin the range of 0.1 to 1% extending down the inner branch of G5, as observedexperimentally. Similar agreements between model and experimental results werealso observed for 1 µm particles [25].

A significant finding from this direct comparison was that the CS and standardDF models did not effectively predict the deposition of fine respiratory aerosols.The CS model has been widely applied in respiratory dynamics systems for particlesup to approximately 100 nm [20,21,52,53]. However, only a few available studieshave applied a DF model for the simulation of respiratory aerosols [34,54]. Noprevious study has implemented a DF model to determine the inertial depositionof fine respiratory aerosols. Based on the available two-phase flow literature, thestandard DF model should adequately model the deposition of dilute fine respiratory



aerosols based on diffusional and inertial transport mechanisms [32,33]. However,results of this study indicated a significant overprediction of respiratory aerosoldeposition with use of the standard DF formulation. This overestimation was due tosignificant changes in particle slip velocity between the near-wall control volumecenter location and conditions at particle-to-wall contact.

11.6 Evaluation of the DF-VC Model in Realistic Airways

In this section, the DF-VC model was extended to transient conditions and wastested in more complex geometries of the respiratory tract for laminar and tur-bulent flows. Particle transport and deposition were evaluated in a computationalreplica of the TB geometry and an MRI-based nasal model. Numerical results werecompared to existing experimental data in these models. A standard CS model wasalso considered to evaluate the extent to which the DF-VC model captures the effectsof particle inertia. To determine the influence of time effects on particle deposition,steady and transient flow fields were evaluated. These studies are intended to furtherdevelop a highly effective method for the simulation of fine respiratory aerosols inrealistic models of the upper airway.

11.6.1 Tracheobronchial region

A hollow cast of the human TB tree (Figure 11.7a) utilized by Cohen et al. [55] wasdigitally replicated in this study to ensure a direct comparison with experimentaldeposition results. The geometric parameters of the cast were in agreement withpopulation means of a representative average adult male. The original cast used inthe study of Cohen et al. [55] was scanned by a multirow-detector helical CT scanner(GE medical systems, Discovery LS) with the following acquisition parameters:0.7 mm effective slice spacing, 0.65 mm overlap, 1.2 mm pitch, and 512 × 512pixel resolution. The multislice CT images were then imported into MIMICS(Materialise, Ann Arbor, MI) to convert the raw image data into a set of cross-sectional contours that define the solid geometry. Based on these contours, a surfacegeometry was manually constructed in Gambit 2.3 (Ansys, Inc.) (Figure 11.7b andc). Some distal branches in the range of generations G5 and G6 were not retainedin the digital model due to low resolution. Most of the digital model paths extendedfrom the trachea to generation G4 with some paths extending to generations G5and G6. Twenty-three outlets and a total of 44 bronchi were retained in the finalcomputational model (Figure 11.7b). The surface geometry was then imported intoANSYS ICEM 10 (Ansys, Inc.) as an IGES file for meshing. To avoid excessivegrid elements, some minor smoothing of the geometric surface was necessary.

The left–right asymmetry, which is an important feature of the human lung, waspreserved in the TB model. There are three lobes in the right lung and two lobes inthe left. As observed in the TB model, the left main bronchus is approximately 2.5times longer than the right (Figure 11.7b), which is consistent with van Ertbruggenet al. [56]. For conducting airways, the bifurcating pattern is typically asymmetric



(a) TB cast (b) Reconstructed surface

CT scansMIMICSICEM

Left Left

Right Right

Glottis

Back view Front view

Near-wall mesh

Figure 11.7. Development of a tracheobronchial airway model from a replica castof the human upper respiratory tract: (a) the hollow cast utilized byCohen et al. [14] showing the divisions of sub-branch segments. Amechanical larynx was connected to the cast in order to generate cyclicinspiratory flows. (The figure was provided by Dr. Beverly Cohen,NYU School of Medicine.) (b) A surface geometry was constructedbased on CT scans of the cast and software packages MIMICS andGambit. Using ANSYS ICEM 10, the computational mesh was gen-erated and consisted of unstructured tetrahedral control volumes witha very fine near-wall grid of prism elements.

with daughter branches from the same parent often differing both in diameter andlength. Additionally, the spatial orientations of the bifurcating branches are quitevariable, resulting in a lung architecture that is highly out-of-plane.

A critical feature of the physiologically realistic TB model used in this studyis the inclusion of a laryngeal approximation and wedge-shaped glottis, as shownin Figure 11.7b and c. Including a laryngeal approximation provides a better rep-resentation of in vivo conditions and is significant in deposition studies with TBcasts [57]. Studies by Chan et al. [58], Gurman et al. [59], Martonen et al. [60],and recently Xi et al. [35] have highlighted the significance of the laryngeal jet ondownstream deposition in the respiratory tract. The transient breathing conditionsof Cohen et al. [55] were approximated as a sinusoidal function with the format

Q(t) = Qin [1 − cos (2ωt)] ; u(t) = umean [1 − cos (2ωt)] (18a and b)

where ω is the breathing frequency in radians/s.Compared with the idealized bifurcation model, interesting flow phenomena

are observed due to the complex geometry characteristics of the TB model suchas the presence of the larynx, left–right asymmetry, and non-coplanar bifurcations.The laryngeal jet that forms at the glottis aperture extends downstream through the



x

z

y

xz x

y

z

x

y

z

x

y

zx

y

z

R2

L2

L1

T2

200 cm/s

T1360

275

190

105

20

Speed(cm/s)

Qin = 34 L/min; steady state

Trachea

T2: SMIF = 0.10

R1: SMIF = 0.52

R2: SMIF = 0.40

L1: SMIF = 0.66

L2: SMIF = 0.43

T3: SMIF = 0.20

Right lung Left lung

R1y

xz

Figure 11.8. Midplane velocity vectors, contours of velocity magnitude, and in-plane streamlines of secondary motion in the tracheobronchial modelunder steady conditions with an inspiratory flow rate of 34 L/min. Theslices are not to scale.

trachea and is skewed toward the right wall (Figure 11.8). Flow reversal occursin the trachea due to the strong jetting effect. As a result, a large recirculationzone develops near the left tracheal wall, which significantly reduces the cross-sectional area available for expansion of the high-speed flow. Similar results havebeen reported by Corcoran and Chigier [61]. Using Laser Doppler Velocimetry andfluorescent dye, Corcoran and Chigier [61] also observed a skewed laryngeal jet inthe right anterior trachea and a region of reverse flow in the left trachea with a castmodel. The skewed jet feature may have physiological implications that facilitateefficient mixing and deeper penetration of inhaled oxygen into the lung. Conversely,higher particle deposition rates are expected on the right tracheal wall and in theright downstream branches due to impaction.



0

0.2

0.4

0.6

0.8

40 nm 200 nm

ExpCS

steady

CStidal

DF-VCtidal

DF-VCtidal

Was

hout

phas

e

CStidal

Exp

DF-VCsteady

CSsteady

DF-VCsteady

Par

ticle

rel

ease

phas

e

Dep

ositi

on fr

actio

n (%

)

Figure 11.9. Comparison of total deposition fraction as a percentage between theexperimental results of Cohen et al. [14] and model predictions understeady and transient conditions for 40 and 200 nm particles.

For transient inlet conditions of a mean flow rate of 34 L/min, the Womersleynumber (α) ranges from 4.2 to 0.8 in the conducting airways of G0 to G6, indicatingmoderate to small unsteady effects. As expected, transient flow patterns are similarto steady state conditions for a major portion of the breathing cycle (i.e., 0.15Tto 0.85T), resulting in what can be considered as a quasi-steady system [42]. Incontrast, during the transition phase between breathing cycles when the veloc-ity is small, the fluid experiences dramatic changes in both mainstream flow andsecondary motion [35,42]. Similarly, Kabilan et al. [62] examined the flow char-acteristics in a CT-based ovine lung model. Their results suggested that the onsetof the transient phase of the flow might be the main cause of vorticity formation,which plays an important role in gas mixing. Li et al. [63] also reported that particledeposition has a strong dependence on individual phases of the transient waveform.

Comparisons of experimental and numerical values of total deposition in theTB model geometry for 40 and 200 nm particles are illustrated in Figure 11.9.The experimental deposition values were reported by Cohen et al. [55] for cyclicbreathing with a mean flow rate of 34 L/min and a breathing frequency of 20 breathsper minute. The transient numerical deposition fraction is a cumulative value thatincludes both the particle-release phase (2nd cycle) and the following washout phase(3rd and 4th cycles). The first cycle is airflow only, without particles, in order toestablish the transient flow field within the TB geometry. In the case of 40 nm parti-cles, the experimental total deposition rate is 0.7%. Both the CS and DF-VC modelsclosely match the experimental deposition value under steady conditions with a totaldeposition prediction of 0.706 and 0.714%, respectively.The marginally higher totaldeposition value of the DF-VC model indicates a negligible contribution of inertiato total deposition for 40 nm particles. In contrast, both models appear to underes-timate the experimental result for transient conditions, resulting in a percent errorof −48.9% for the transient CS model and −21.1% for the transient DF-VC model.



1.8

0.9

0

2.7

3.6DEF

Transientventilation

DEFmax = 17.67

Transientventilation

DEFmax = 266.77

(a) 40 nm (b) 200 nm

RightLeft

1.8

0.9

0

2.7

3.6

DEF

RightLeft

Figure 11.10. Numerically determined deposition enhancement factors (DEF) inthe tracheobronchial airway under transient conditions at a mean flowrate of 34 L/min with the DF-VC model for (a) 40 nm and (b) 200 nmparticles.

Experimental cyclic deposition results for 200 nm particles indicate a total depo-sition fraction of 0.38% (Figure 11.9). Due to a lack of particle inertia, the CSmodel significantly under predicts the total deposition fraction for 200 nm aerosols,resulting in percent errors of −52.1% for steady ventilation and −81.3% for cyclicventilation. The DF-VC model provides a substantial improvement over the CSapproach by incorporating the particle inertia forces in the computational model.As with 40 nm particles, predictions of the DF-VC model for 200 nm particles withsteady flow are in close agreement with the experimental results, with a minorunderestimation of −8.4%. For cyclic conditions, the DF-VC approach predicts atotal deposition of 0.257%, resulting in a difference of −32.4% in comparison withthe experiment. Comparing these two models, differences between the CS predic-tions and the experimental results are significant for 200 nm particles, indicatinga significant contribution from particle inertia. Still, an under prediction of depo-sition for 200 nm particles is observed with the DF-VC model. As noted before,some distal branches of the TB cast were not included in the numerical model anda static open glottis was employed, which may contribute to this under predictionof the experimental results for both steady and transient inhalation conditions.

Figure 11.10 illustrates the DEF value for the DF-VC model under cyclic con-ditions with a mean flow rate of 34 L/min. For the DF-VC model, the depositionpatterns under steady and cyclic conditions are similar in appearance for 40 and200 nm particles, respectively. However, more concentrated local deposition valuesin the larynx and bifurcations are obtained under cyclic conditions compared withsteady state. Specifically, the maximum DEF value for 200 nm aerosols with cyclicflow (i.e., 267) is about seven times higher than with the steady condition (i.e., 37),



Nasopharynx Main passage

IM

LP

MMUP

OR

SM

Valveregion

Vestibule

Figure 11.11. Surface geometry of a MRI-based human nasal cavity. The nasalairway was divided into different anatomical sections to exam-ine sub-regional particle deposition. Sections include the vestibule,nasal valve region, olfactory region (OR), septum wall (SW), middlemeatus (MM), as well as the upper (UP) and lower (LP) passagesthat include the superior (SM) and inferior (IM) meatus, respectively,and the nasopharynx.

indicating a dramatic increase of the velocity-dependent inertia effect on particlelocalization. For the smaller 40 nm particles, the maximum DEF value of approxi-mately 18 for cyclic inhalation is only twice as large as the steady state value of 8.3.As a result, transient conditions have a greater effect on the deposition of a 200 nmaerosol, in comparison with 40 nm particles, due to inertial effects.

Geometric complexity, turbulence, and transient flow are all expected to signif-icantly influence the deposition of aerosols in the respiratory tract [38,64–66]. Inthis context, an extended DF-VC model was evaluated by comparing with exper-imental results in a complex TB geometry with transient and laminar-to-turbulentflows. Through comparison with the CS model, the DF-VC approach was shownto better capture the expected effects of particle inertia on a total and local basis.

11.6.2 Nasal cavity

The DF-VC model was further tested in a human nasal airway, which was basedon MRI scans of a healthy non-smoking 53-year-old male (weight 73 kg and height173 cm). The procedure for developing the nasal airway geometry and mesh issimilar to the approach applied for theTB model in the previous section.As shown inFigure 11.11, the nasal cavity is characterized by narrow, convoluted, and multilayerchannels called meatus. This complex passageway generates unique aerodynamicsinside the nasal cavity and helps the nose to accomplish its physiological functions.Conversely, reasons for deficient nasal function can often be traced back to abnormal



(b) DF-VC model

DE = 0.021%DEF DEF

DE = 0.067%

0 1 2 3 4 50 1 2 3 4 5

DEFmax = 97 DEFmax = 592

(a) CS model

Qin = 30 L/mindp = 400 nm

Figure 11.12. Numerically determined deposition enhancement factors (DEF) inthe nasal airway for 400 nm particles at an inhalation flow rate of30 L/min using the (a) CS and (b) DF-VC models.

global and local flow conditions and turbulent phenomena in the nasal passages. Inan adult, 18,000 to 20,000 L of air pass through the nose each day. Besides warmingand moistening the inhaled ambient air, the nasal cavity also houses olfactorysensory receptors, filters out airborne pollutants, drains excess sinus secretions,and balances pressure between the middle ear and atmosphere.

As in the previous section, two continuous field particle transport models, i.e.,CS and DF-VC models, are considered. To highlight the effect of finite particleinertia on deposition localization, a comparison of DEFs predicted using the CSand DF-VC models are shown in Figure 11.12 for 400 nm particles at an inhalationflow rate of 30 L/min. With DEF values plotted on the same scale, the differencebetween these two models is striking. In contrast to the more uniformly distributedDEF values of the CS model, the deposition with the DF-VC model is significantlymore heterogeneous and localized. Specifically, at the anterior junction point (solidcircle) between the middle meatus and medial passage where high-speed flow andsteep geometry transition occur, the hot spot of the DEF value predicted by theDF-VC model is about five times higher than the CS value. Another expected hotspot is at the superior part of the vestibule (dashed square), which is capturedby the DF-VC model. In contrast, the CS model does not indicate this region ashaving elevated deposition. Additionally, elevated deposition accumulations arealso predicted around the rear olfactory region (filled arrow) based on the DF-VCmodel due to trajectory deviations of large particles from curved streamlines inthe main airflow. These elevated localizations may have important implicationsin chemical sensing applications or nasal drug delivery for neurological disorderswhere the olfactory region is the targeted deposition site.

The effectiveness of the DF-VC model in capturing inertial and diffusive depo-sition is further illustrated in Figure 11.13 in terms of deposition within specificsections of the nose. The definition of each section is depicted in Figure 11.11



0

LPMMUP

OR

SW

LP

MM

SW

OR

0.015

0.01

0.005

Dep

ositi

on fr

actio

n (%

)

Vestibule Nasal valveregion

Main passage Nasopharynx

Rig

ht

DF-VC model

CS model

Qin = 30L /min, dp = 400 nm

Left

UP

Figure 11.13. Comparison of deposition fractions between the CS and DF-VCmodels in different nasal sections. The magnitude of the depositionfraction values in the anterior and main nasal airway is the summationof the right (lower bar) and left (upper bar) nasal passages.

and the inhalation conditions are identical to those shown in Figure 11.12(i.e., Qin = 30 L/min and 400 nm particles). The magnitude of the deposition frac-tion value for each region (except for the nasopharynx) represents the summationof right (lower bar) and left (upper bar) nasal passages. The deposition fractionvalues for the CS model denote deposition from diffusion only, while the differ-ence between the DF-VC and CS models can be viewed as deposition from inertialimpaction. In the vestibule, where the direction of airflow changes by approxi-mately 90, significantly enhanced deposition is predicted by the DF-VC model,which is about 6.1 times the CS model estimate for identical conditions. An evenmore pronounced effect of inertial impaction (i.e., an order of magnitude increase)is found in the nasal valve region where the cross-sectional area is minimal and theairflow passages are narrowest. In this region, the valve-associated stenosis, flowacceleration, and short distances to the walls combine to effectively increase theinertial deposition of particles entrained in the flow. The inertial effect on depositionin the main passage remains significant, but to a lesser degree in comparison withthe vestibule and nasal valve regions, which is likely due to the increased flow area(decreased flow speed) and less severe streamline curvatures. Within the main pas-sage, diffusion is enhanced in the middle and inferior meatus by the slow-movingflows (and the resulting prolonged particle residence times) as well as the largesurface areas of these two fin-like projections available for particle contact. As aresult, the deposition fraction of the DF-VC model is only 3.0 times that of the CSmodel for the middle meatus and 2.9 times larger for the lower passage.

Figure 11.14a shows the DF-VC simulation results of inspiratory depositionfractions in comparison to replica measurements as a function of a diffusion



X

XXXXXX

XX

XXXXX

XX

XXXXXX

XXX

XX

XXX

XXXX

XXXXXX

XXX

XXXXXX

X0

20

40

60

80 XSLA (Kelly et al., 2004)Viper (Kelly et al., 2004)ANOT1 (Cheng et al., 1995)ANOT2 (Cheng et al., 1995)DF-VC simulation

Dep

ositi

on fr

actio

n (%

)

100

10−3 10−2 10−1

(a) Inhalation diffusion parameter (D1/2Q−1/8)

C1 (Cheng et al., 1993)ANOT1 (Cheng et al., 1995)ANOT2 (Cheng et al., 1995)DF-VC model

Dep

ositi

on fr

actio

n (%

)

0

20

40

60

80

100

10−310−4 10−2 10−1

(b) Exhalation diffusion parameter (D 0.5Q −0.28)

Figure 11.14. Predicted deposition rates with the DF-VC model in comparisonwith existing in vitro experimental data for particles ranging from 1to 1,000 nm and inhalation flow rates ranging from 4 to 45 L/min:(a) inspiratory, (b) expiratory. Units: Q [L/min] and D [cm2/s].

parameter suggested by Cheng et al. [67]. Inhalation flow rates that were con-sidered included 4, 7.5, 10, 15, 20, and 30 L/min with particle sizes ranging from1 to 1,000 nm. The correlated association of deposition fraction with the parameter(D1/2Q−1/8) indicates a stronger dependence of nasal deposition on particle diffu-sivity (exponent of 1/2) than on flow rate (exponent − 1/8) for the submicrometerparticles considered.

Figure 11.14b shows the DF-VC simulation results of expiratory depositionfractions in comparison with existing in vitro deposition data. Simulated expira-tory deposition rates with the discrete Lagrangian tracking is also superimposedin this figure, which closely matches the continuous-phase DF-VC results. FromFigure 11.14b, reasonably good agreement is observed between simulations and invitro measurements for the three replica casts considered with slight underestima-tion to the in vitro data. Specifically, the closely correlated association of depositionfraction with the parameter (D0.5Q−0.28) indicates a stronger dependence of nasaldeposition on particle diffusivity (exponent of 0.5) than on flow rate (exponent−0.28) for the submicrometer particles considered.

As described earlier, the DF-VC approximation accounts for both particle inertiaand diffusion, whereas the CS model only considers particle diffusion. As a result,the increased deposition of the DF-VC model over the CS approach can be attributedentirely to particle inertial effects. For ultrafine particles where inertial effectsare negligible, the Sherwood number Sh is nearly identical for the two modelsconsidered (Figure 11.15a). Deviation of Sh between the DF-VC and CS modelsbegins at Re0.55Sc0.60 = 1.5 × 104 (equivalently, Stk = 1.0 × 10−5) and becomesprogressively significant with increasing values of Re0.55Sc0.60 (or Stk). Comparedwith the inertial limit of Stk = 5.0 × 10−5 for TB airways [20], a smaller value in



R 2 = 0.80

DSh = 4.7 × 106 Stk1.1

100

101

102

103

104

She

rwoo

d nu

mbe

r de

viat

ion,

∆S

h

DSh

(a) Correlation for diffusion regime (b) Correlation for inertia-based deposition

Stk

10−310−410−5102 103 104 105

Re0.55Sc0.60

Sh = (Re0.55Sc0.60)0.553

R 2 = 0.93

CS modelDF-VC modelCorrelation for diffusion

She

rwoo

d nu

mbe

r, S

h

104

103

102

101

Diffusion- impactionDiffusion

Sh deviationCorrection for inertia

Figure 11.15. Development of a Sherwood number Sh correlation in the nasalairway that accounts for both diffusional and inertial depositionmechanisms for submicrometer aerosols: (a) correlation for ultrafineparticles where diffusion dominates deposition and (b) correlationfor enhanced mass transport due to particle inertia,Sh, as a functionof Stokes number, Stk.

the nasal cavity indicates an earlier onset of particle inertial effects, which may bedue to the high complexity of this geometry.

While the nasal deposition can be reasonably predicted by existent empiricalcorrelations in the literature, these expressions are typically limited to ultrafine orcoarse particles where either diffusion or inertial impaction are the predominatemechanisms for particle loss. Therefore, a correlation that is valid for both diffu-sional and inertial deposition regimes is of value to inhalation toxicologists andis sought in this study based on numerical experiments. Figure 11.15 shows thedevelopment of a mass transfer correlation in the nasal cavity in terms of the non-dimensional Sherwood number. The dependence of mass transfer on convectivediffusion is plotted in Figure 11.15a with a best-fit correlation for the diffusionzone given by

Sh = (Re0.55Sc0.60)0.553 = Re0.30Sc0.33 (Stk ≤ 1.0 × 10−5) (19)

The convective–diffusion coefficient (Re0.55Sc0.60) adopted here was suggested byCheng et al. [68] for in vivo nasal deposition data. For fine respiratory particles thatare influenced by diffusion and impaction deposition mechanisms, the deviationfrom the above equation (Sh, see Figure 11.15a) due to particle inertia can becorrelated as a function of Stk using (Figure 11.15b)

Sh = 4.7 × 106 St1.1k (20)



Based on the assumption of weak coupling between the inertial and diffusive deposi-tion mechanisms, the overall correlation that is valid for all submicrometer particlesizes can be obtained by adding equations (19) and (20) as:

Sh = Re0.30Sc0.33 + 4.7 × 106St1.1k (dp = 1 − 1, 000 nm) (21)

Alternatively, the above correlation can be expressed in terms of total nasaldeposition fraction,

Df = 1 − exp

[−

(As

Ac

)(Re−0.70Sc−0.67 + 4.7 × 106 St1.1k

Re Sc

)](22)

where As is the total surface area and Ac the mean cross-sectional area of the nasalpassage. The above equation is valid for particle size ranging from 1 to 1,000 nm.

11.7 Discussion

Numerical simulations of fine respiratory aerosols are challenging due to low depo-sition efficiencies and the action of concurrent inertial and diffusive depositionmechanisms. The CS Eulerian model that is typically applied for ultrafine aerosolsis highly efficient and shows reasonable agreement with analytic solutions for ultra-fine diffusive deposition in a tubular geometry [17,20,69]. However, the typical CSEulerian model neglects the effects of finite particle inertia [20,32] and cannot beused to predict fine particles in the range of 100 to 1,000 nm. As presented in thischapter, finite particle inertia may significantly affect the total deposition of submi-crometer aerosols as small as 70 nm and the local deposition of particles as small as40 nm [25,40,42]. Based on the available two-phase flow literature, the standard DFmodel should adequately model the deposition of dilute fine respiratory aerosolsbased on diffusional and inertial transport mechanisms [32,33]. However, resultspresented by Longest and Oldham [25] and summarized above indicated a signif-icant overprediction of respiratory aerosol deposition with the use of the standardDF formulation. This overestimation was due to significant changes in particle slipvelocity between the near-wall control volume center location and conditions atparticle-to-wall contact. To improve the simulations of submicrometer aerosols inthe respiratory tract, a hybrid DF-VC model was developed and tested both experi-mentally and numerically [25,40,42]. In comparison with the CS model, depositionresults of the DF-VC approach persistently agreed better with experimental findingson a total and subbranch basis, which indicated that the DF-VC model effectivelycaptured the influence of finite particle inertia. Results of this study indicate thata DF particle transport model with near-wall velocity corrections can provide aneffective approach for simulating the transport and deposition of submicrometerrespiratory aerosols in complex human respiratory airways.

While the DF-VC model has been shown to be effective for submicrometer res-piratory aerosols, it does have several limitations. The current DF-VC model was



developed mainly for submicrometer aerosols based on the use of the pressure gra-dient term in equation (11b). This model can also be extended to larger particle sizesif equation (11a) is used to compute particle slip. Another limitation of the generalDF approach is the assumption of a dilute aerosol in which the momentum of thediscrete phase does not influence the flow field. As a result, low loadings of parti-cle concentration are required, which is generally the case for respiratory aerosols.Finally, the current model was developed for monodisperse aerosol distributions.

The results of the studies summarized in this chapter should not be interpretedto imply that the continuous field DF-VC model is superior to Lagrangian parti-cle tracking for simulating aerosol transport in all situations. A primary advantageof Lagrangian particle tracking is the direct treatment of particles as individualor potentially interacting discrete elements, in contrast with the approximation ofa continuous aerosol field. Treating particles as discrete elements allows for thedirect evaluation of individual force terms that represent a number of physicalphenomena such as lift, near-wall lubrication, and particle-to-particle interactions[29]. Furthermore, Lagrangian particle tracking can be directly applied to non-spherical shapes, like rotating fibers, and polydisperse aerosols. In contrast to theLagrangian approach, the DF-VC model approximates finite particles as a continu-ous species. As a result, transport and deposition are evaluated from the solution ofa convection–diffusion equation, which may introduce numerical dissipation errorsinto the solution. Furthermore, it does not appear that the DF-VC model can effec-tively account for particle-to-particle interactions or the motion and interceptionof complex shapes, like rotating fibers. A primary advantage of the DF-VC modelis high efficiency, where successive additions of discrete particles are not requiredto establish converged local deposition characteristics. As a result of these obser-vations, it is clear that both the continuous field DF-VC model and Lagrangianparticle tracking have inherent strengths and weaknesses. Further testing and eval-uation is required to extend the flexibility of the DF-VC model and to overcomesome current limitations, such as monodisperse aerosols. It is likely that selection ofEulerian versus Lagrangian models for submicrometer aerosol transport will remaindependent on the particular problem of interest in conjunction with the advantagesand disadvantages of each method. However, results of this study indicate that theDF-VC model does provide an effective approach that can simulate the concur-rent action of diffusion and inertia in a complex transient and laminar-to-turbulentflow field.

In conclusion, a continuous-phase DF particle transport model with a near-wallvelocity correction provided an effective solution for the deposition of submicrome-ter aerosols under transient and laminar-to-turbulent flow conditions. Comparisonswith a standard CS model indicated that the DF-VC approach was incorporating theeffects of finite particle inertia in determining total and local deposition character-istics for the complex TB and nasal geometries that were considered. Comparisonsof steady and transient conditions indicated that cyclical flow decreased the totaldeposition of submicrometer particles but significantly increased relative particlelocalization, as indicated with the DEF parameter. Future studies are necessaryto address wall surface roughness and compliance, mucus clearance, and real-life



breathing waveforms before the current DF model can be directly applied to makedose–response and health effects predictions.

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12 Algorithm stabilization and acceleration incomputational fluid dynamics: exploitingrecursive properties of fixed pointalgorithms

Aleksandar Jemcov and Joseph P. MaruszewskiANSYS, Inc., Canonsburg, PA, USA

Abstract

Convergence acceleration of nonlinear flow solvers through use of techniques thatexploit recursive properties fixed-point methods of computational fluid dynam-ics (CFD) algorithms is presented. It is shown that nonlinear iterative algorithmscreate Krylov subspace that can be considered linear within the second orderof accuracy in the vicinity of the local solutions. Moreover, repeated applica-tion of the fixed-point iterations give rise to various vector extrapolation methodsthat are used to accelerate underlying iterative method. In particular, suitabilityof the reduced rank extrapolation (RRE) algorithm for the use of convergenceacceleration of nonlinear flow solvers is examined. In the RRE algorithm, thesolution is obtained through a linear combination of Krylov vectors with weight-ing coefficients obtained by minimizing L2 norm of error in this space withproperly chosen constraint conditions. This process effectively defines vectorsequence extrapolation process in Krylov subspace that corresponds to the GMRESmethod applied to nonlinear problems. Moreover, when the RRE algorithm isused to solve nonlinear problems, the flow solver plays the role of the precon-ditioner for the nonlinear GMRES method. Benefits of the application of theRRE algorithm include better convergence rates, removal of residual stalling, andimproved coupling between equations in numerical models. Proposed algorithmis independent of the type of flow solver and it is equally applicable to explicit-,implicit-, pressure-, and density-, based algorithms. RRE algorithm can also beconsidered as an generalization of matrix-free algorithms that are used extensivelyin CFD.

Keywords: Fixed-point methods, Krylov subspace, Recursive properties, Reducedrank extrapolation, Vector extrapolation



12.1 Introduction

Computational fluid dynamics (CFD) in modern engineering applications is char-acterized by complex physical models, complex shapes of computational domain,and large mesh sizes. Governing equations of fluid flow and associated equationsfor the transport of conserved variables are seemingly simple and yet they containvery prominent nonlinearities of which the most famous one is manifested throughthe phenomena of turbulence. The need to model turbulent flows coupled withradiation, heat and mass transfer, and multiphase interactions lead to very complexphysical models that contain many nonlinear terms and wide range of spatial andtemporal scales. This complexity directly translates into difficulties in numericalapplications and they are loosely referred to here as computational stiffness (forthe current purposes of the qualitative discussion, we will use the term stiffnessto signify computational difficulties). Another source of stiffness in CFD is asso-ciated with the use of large mesh sizes in order to capture relevant spatial scalesassociated with physical models. It is well-known fact that the condition numberof linear system increases with the increase of the mesh size as it can be verifiedby the uniform refinement of the computational mesh. Furthermore, topologicalcomplexities of the computational domains lead CFD practitioners to the use ofunstructured meshes with nearly arbitrary shapes of control volumes. The mainissues of unstructured meshes such as presence of highly skewed volumes, highaspect ratios close to the boundaries, loss of orthogonality, and lack of convexitydirectly contribute to the stiffness of the computational problem. This is usuallymanifested in the loss of the diagonal dominance of the matrix in the linear systemof discrete equations. Iterative solvers for linear systems of equations are not par-ticularly effective when the matrix of the linear system departs strongly from thediagonal dominance expressed through M -matrix condition.

Complexity is not just restricted to physical models and related issues asdescribed above; flow simulation is also characterized by a great number of dif-fering algorithms ranging from explicit to fractional step to implicit methods [1].Furthermore, depending on the choice of the solution variables and chosen discreteformulation, pressure [2–4] and density-based solvers [5,6] can be distinguished.Pressure- and density-based solvers can be formulated as implicit, explicit, or frac-tional step methods. Complexity of the landscape is further increased by the choiceof linear solver used in implicit algorithms. Additionally, various flux formulationsare in the use today including Riemann solver, flux vector splitting, central dif-ference with dissipation flux formulation [6], to name just a few. Each of thesealgorithms have control parameters that have optimal values for a given applica-tion area thus producing an optimal convergence rate during the iterative process.Examples of control parameters are CFL numbers in density-based and relaxationfactors in pressure-based algorithms.

Optimal range of control parameters are defined by the stability constraintsof the numerical algorithm, underlying physics, computational mesh, geometricalcomplexity of the computational domain, and boundary conditions. Due to the non-linear nature of equations describing the flow phenomena, selection of the values


exploiting recursive properties of fixed point algorithms 461

of control parameters that are outside of the optimal range results in either slowconvergence rates or divergence of the residual norm. Linear stability analysis [5]provides some guidelines for the selection of the optimal values of control param-eters but in the industrial applications of flow solvers, problem complexity dueto physics, domain geometry, boundary conditions, etc. dramatically changes theoptimal range of control parameters. Furthermore, the nonlinear nature of govern-ing equations changes the optimal values of control parameters during the iterativeprocess. During initial iterations, depending on the initial guess, optimal values arevery different from the optimal values of the control parameters during later stagesin iterative process. Usually, initially conservative values of control parameters areused to maintain the stability of the iterative process and as iterations approach thesolution, more aggressive values can be used.

The need to dynamically change the values of control parameters during itera-tions is addressed in algorithms that compute optimal or nearly optimal values byutilizing the knowledge of the state of computed variables and some criterion thatmeasures stability of the computed state. Continuation methods [7] provide a pos-sible solution to this problem. Another approach to the problem of optimal controlparameters is in the use of backtracking or line search algorithm in conjunction withNewton methods [8]. Both continuation and Newton with backtracking algorithmsare effective tools for the solution of the problem of dynamically changing controlparameters but they require algorithmic changes to the flow solver. Many times it isnot feasible to implement new algorithms in large codes, and even though the newalgorithms provide added stability and increase convergence rates, they cannot beimplemented in timely manner. Less intrusive ways of improving the convergencerates and overall stability of the flow solver are required.

Another source of instability and low convergence rates in flow solvers is relatedto artificial segregation of discrete equations in the numerical model. Example ofthe algorithms that use segregated approach are the SIMPLE family of algorithms[2,4]. Another example of segregated algorithms is found in the treatment of tur-bulence models [9] in density-based algorithms where they are lagged behind flowquantities. In multi-equation turbulence models, each turbulent transport equationmay be solved separately, thus artificially decoupling equations in the numericalmodel. Decoupling of equations in segregated flow solvers has a detrimental impacton convergence rates. The solution to this problem is usually found in creation ofcoupled solvers at the expense of increased memory and code complexity.

Vector sequence extrapolation methods [10] offer an alternative solution to con-vergence acceleration and stabilization of flow solvers. The idea of vector sequenceextrapolation is based on the generalization of scalar sequence convergence accel-eration methods, such as Richardson extrapolation [10], to vector sequences. Vectorsequences are produced by iterative procedures of solver algorithms while seekingthe solution of the given flow problem. In addition to accelerating properties ofvector sequence extrapolation methods, they also exhibit coupling properties sinceonly one set of extrapolation coefficients is calculated for all fields. This has apositive effect on convergence rates, very similar to the action of Krylov subspacesolvers acting on systems with block coefficients. Moreover, formal equivalence



between vector sequence extrapolation methods to Krylov subspace solvers suchas GMRES and FOM [11] exists.

Early examples of the application [13] of vector sequence methods in CFDwas the application of reduced rank extrapolation (RRE), minimal polynomialextrapolation (MPE), and modified minimal polynomial extrapolation (MMPE)to explicit flow solvers based on JST scheme [12] and implicit flow solver with fluxvector splitting formulation. Acceleration of convergence rates of inviscid flowsolvers was examined [13] and marginal improvements were obtained with explicitsolvers, whereas implicit solvers showed better results. Another application [14]of an extrapolation-related technique to several types of flow demonstrated thesuitability of extrapolation to accelerate inviscid compressible and viscous laminarincompressible flows. The method [14] used in the studies relied on residual mini-mization and required many evaluations of residuals making this method expensive.Other applications of vector sequence extrapolation to nonlinear problems includeChandrasekhar’s H-equation [15]. Vector sequence extrapolation is also used foracceleration of stationary iterative methods [16] and the algebraic multigrid (AMG)method [17].

Traditional view of extrapolation methods applied to vector sequences is thatthey perform the transformation of one sequence into another sequence result-ing with faster convergence rates. Since a flow solver will be generating vectorsequences that will be used to compute extrapolation coefficients, an alternativeview of this process is that the flow solver plays the role of a nonlinear preconditionerfor an extrapolation solver. Due to equivalence of RRE and GMRES algorithms,the proposed acceleration technique corresponds to a nonlinearly preconditionedGMRES solver.

The outline of the paper is as follows. CFD algorithms were examined from thefixed-point algorithm point of view and the recursive property of these algorithms isshown. Recursive property of the fixed-point algorithms represent, to second-orderapproximation, a power iteration with the matrix of Jacobian of the fixed-pointfunction. This property plays an important role in the derivation of vector sequenceextrapolation algorithms. This is followed by a description of the vector sequenceextrapolation method where the recursive property of the fixed-point function wasused to obtain the extrapolation coefficients. This is done through the formulation ofan optimization problem in the Krylov subspace and a proper choice of constraintslead to the RRE algorithm. In the section on numerical experiments, the applicationof nonlinear acceleration is demonstrated on various CFD problems utilizing severalflow solvers. Finally, a summary and conclusion is presented, following results ofnumerical experiments.

12.2 Iterative Methods for Flow Equations

In this section we will consider the discrete form of the governing equations anditerative methods of finding the solution for nonlinear problems in CFD. Iterativemethods are considered from the point of view of the fixed-point algorithms that



have a recursive property related to powers of the Jacobian of the fixed-point func-tion. Recursive property plays important role in establishing the existence of Krylovsubspace in which the solution of the problem is approximated. In fact, we demon-strate how the recursive property generates Krylov subspace for any fixed-pointfunction thus enabling acceleration and stabilization of arbitrary iterative problemsunder the certain conditions.

12.2.1 Discrete form of governing equations

Consider Navier–Stokes system of equations in integral form

∂

∂t

∫

Qd+∮∂

(Fc − Fv)dA −∫

Sd = 0 (1)

Here Q represents vector of conserved quantities, Fc and Fv are convective andviscous flux vectors, and S is the source term vector

Q =

⎛⎜⎜⎜⎜⎝ρ

ρuρv

ρwρE

⎞⎟⎟⎟⎟⎠ , Fc =

⎛⎜⎜⎜⎜⎝ρV

ρuV + nxpρvV + nypρwV + nzpρHV

⎞⎟⎟⎟⎟⎠ , Fv =

⎛⎜⎜⎜⎜⎝0

nxτxx + nyτxy + nzτxznxτyx + nyτyy + nzτyz

nxτzx + nyτzy + nzτzznx x + ny y + nz z

⎞⎟⎟⎟⎟⎠

S =

⎛⎜⎜⎜⎜⎝0ρfe,xρfe,yρfe,z

ρ(fe,xu + fe,yv+ fe,zw) + qh

⎞⎟⎟⎟⎟⎠and V and x, y, and z are defined as:

V = nxu + nyv+ nzw

x = uτxx + vτxy + wτxz + k∂T

∂x

y = uτyx + vτyy + wτyz + k∂T

∂y

z = uτzx + vτzy + wτzz + k∂T

∂z

The system of equations in equation (1) may include additional scalar or vectortransport equations

∂

∂t

∫

d+∮∂

(F,c − F,v

)dA −

∫

Sd = 0 (2)



Depending on the choice of the transported quantity, equation (2) can representspecies, passive scalar, turbulent transport, etc. Appropriate boundary conditions[1] must be also supplied to define a well-posed problem that can be solvednumerically.

The first step in obtaining the discretized equations is to define a residual R(Q)

R(Q) =∮∂

(Fc − Fv) dA −∫

Sd (3)

Equation (3) defines a continuous residual that can be discretized on a computationalmesh using a cell-centered finite volume method. For ith finite volume cell, discreteresidual is given by:

Ri(Q) =∑

f

(F fc − F f

v )i − Si (4)

Here symbol∑

f ( · ) represents summation over all faces of a given finite volumecell i. If the discrete value of the conserved variables vector Qi is defined as:

Qi =∫i

Qd

i(5)

The discrete representation of the Navier–Stokes system in equation (1) becomes

d (Q)idt

= −Ri(Q) (6)

The discrete system of equations given by equation (6) represents the nonlineardiscrete system due to nonlinear nature of the residual Ri. An explicit algorithmcan use the system in equation (6) directly by applying the time marching proceduresuch as the multistage Runge–Kutta [1] procedure and full approximation scheme(FAS) [18] to obtain the solution of the problem. The explicit algorithm is given bythe following expression:

Q(ν+1)i = − t

()iRi(Q(ν)) (7)

Q(ν+1) = Q(ν) + Q(ν)

In the case of an implicit algorithm, a Newton or Picard linearization of the resid-ual is applied to form an iterative scheme. If Newton linearization is used, thediscretized system of equations is given by the following expression:[

()i

t+ (∂QR)i

]Q(ν)

i = −Ri(Q(ν)) (8)

Q(ν+1) = Q(ν) + Q(ν)



Since the solution of the system in equation (8) requires inversion of matrix onthe left-hand side, iterative methods for solving large sparse systems of linearequations are used. Popular choices for linear solvers are Krylov subspace methods[11] and AMG method [18]. There are also number of choices for the convectiveand viscous flux evaluations at cell faces, popular choices in density-based solversare flux difference splitting and flux vector splitting [1], whereas in the case ofpressure-based solvers, collocated schemes with Rhie–Chow dissipation [4] areoften used.

12.2.2 Recursive property of iterative methods

Both explicit and implicit algorithms are particular instances of the fixed-pointmethod that can be generally expressed as:

Q(ν+1) = Q(ν) − M(ν)(R(Q(ν))) (9)

The general form of any fixed-point method is

Q(ν+1) = F(Q(ν)) (10)

with fixed-point function F(Q(ν))

F(Q(ν)) = Q(ν) − M(ν)(R(Q(ν))) (11)

Operator M(ν)(R(Q(ν))) is, in principle, a nonlinear operator that may not evenbe known explicitly. In the case of stationary linear iteration schemes such asGauss–Seidel, Jacobi, and ILU(n), this operator may be constructed explicitly.However, in the case of the AMG method with multiple coarse levels, it is usuallyknown implicitly through implementation in the flow solver. In the case of nonlinearflow solvers, this operator is almost always defined algorithmically through codeimplementation except in trivial cases. However, as it will be shown later, thisknowledge is not required for the vector sequence extrapolation method to work.

An important property of any fixed-point algorithm is the recursive relation thatcan be established between the initial solution correctionQ(0) and later correctionsQ(ν) through powers of the fixed-point function Jacobian ∂QF. This relation canbe established by forming the following difference:

Q(ν) = Q(ν+1) − Q(ν) (12)

From the definition of the fixed-point algorithm, equation (10), equation (12)becomes

Q(ν) = F(Q(ν)) − F(Q(ν−1)) (13)

By the definition, Q(ν) is given by

Q(ν) = Q(ν−1) + Q(ν−1) (14)



and if this expression is substituted in equation (13), we arrive to expression

Q(ν) = F(Q(ν−1) + Q(ν−1)) − F(Q(ν−1)) (15)

Using the linear expansion of the term F(Q(ν−1) + Q(ν−1)) in the previousequation, we obtain

F(Q(ν−1) + Q(ν−1)) = F(Q(ν−1)) + ∂QF(Q(ν−1)) + O(ε2) (16)

Neglecting the terms of higher order, expression in equation (16) becomes

F(Q(ν−1) + Q(ν−1)) ≈ F(Q(ν−1)) + ∂QF(Q(ν−1)) (17)

Substituting equation (17) into equation (15), expression linking Q(ν) andQ(ν−1) is formulated

Q(ν) ≈ ∂QF(Q(ν−1))Q(ν−1) (18)

By letting the index ν to range from 0 to ν+ 1, equation (18) is used recursively toobtain the link between Q(0) and Q(ν+1)

Q(ν+1) ≈ (∂QF(Q(ν)))νQ(0) (19)

The recursive property of the fixed-point algorithm is given by equation (19) andupon closer inspection it can be seen that it represents, a power iteration of Q(ν+1)

with matrix ∂QF(Q(ν)), within second order of accuracy. Moreover, it can also beobserved that the sequence of Q(ν+1) vectors will tend to lie in the dominanteigenspace of matrix ∂QF(Q(ν)) and also define the basis of the Krylov subspaceassociated with the iterative procedure described by the fixed-point method, equa-tion (9). This property is used to approximate the solution of the fixed-point problemby using the minimal polynomial of the Jacobian matrix ∂QF(Q(ν)). A Krylov sub-space generated by the iterative process defined by equation (10) can be definedformally by using the local linearization of the fixed-point function

Kn(M−1(∂QR), R(0)) = span(R(0), (M−1(∂QR))R(0), . . . , (M−1(∂QR))n−1R(0))(20)

Since in equation (20) operator M is kept constant in the local neighborhood ofsolution vectors, Krylov subspace Kn can be shown to be equivalent [14,15] to thespace spanned by correction vectors Q(ν)

Kn(M−1(∂QR), R(0)) = M−1Kn(Q(n))(21)= M−1span(Q(0), Q(1), · · · , Q(n))

For strongly nonlinear fixed-point function, the local approximation used in equa-tion (21) is not valid and the space generated by corrections Q(ν) is no longer



equivalent to the classical Krylov subspace. However, the solution to the fixed-point problem can be still sought in this space although convergence cannot beguaranteed.

12.3 Reduced Rank Extrapolation

After n iterations, vector sequence extrapolation searches for the solution of thefixed-point problem, equation (10), in the space defined in expression equation (21)by approximating the solution by the following linear combination:

Q(n) = Q(n−1) +n∑ν=0

ανe(ν) (22)

Here αν are extrapolation coefficients that need to be determined, e is error after niterations defined as

e(ν) = Q(ν) − Q∗ (23)

and Q∗ is the solution (fixed point) of the problem. Error e(ν) shares the recursiveproperty defined in equation (19)

e(ν+1) ≈ (∂QF(Q(ν)))νe(0) (24)

Equation (24) represents the approximate error propagation equation of the givenfixed-point function and it follows from the linearity of the recursive relation, equa-tion (19). The recursive property of the error, equation (24), is used to defineextrapolation coefficients. To achieve that, first we must find what conditions errormust satisfy.

At convergence, the following identity holds

Q∗ = Q∗ +n∑ν=0

ανe(ν) (25)

due to the fact that the error e is zero. Therefore, at convergence after n iterations,the following identity must be true:

n∑ν=0

ανe(ν) = 0 (26)

Equation (26) represents the formal condition that can be used to determine extrap-olation coefficients αν. Before coefficients can be computed, a formal connectionbetween equation (26) and Krylov subspace Kn(Q(n)) must be established.

It is a known fact that Krylov space methods for linear problems have theproperty to determine implicitly a minimal polynomial of the system matrix with



respect to initial residual. If the minimal polynomial of the Jacobian of the fixed-point function is denoted by Pm(∂QF), then the following identity will hold locally

Pm(∂QF)Q(0) =n∑ν=0

cν(∂QF(Q(ν)))νQ(0) = 0 (27)

Furthermore, using equations (24) and (27)

n∑ν=0

cν(∂QF(Q(ν)))νQ(0) = (I − ∂QF(Q(ν)))n∑ν=0

cνe(ν) = 0 (28)

and since (I − ∂QF(Q(ν))) has a full rank we obtain the following expression:

n∑ν=0

cνe(ν) = 0 (29)

From this analysis it is possible to conclude that if the coefficients cν are chosen sothat they are the coefficients of the minimal polynomial Pm with respect to Q(0),then the condition in equation (29) holds. If we chose coefficients αν to be thecoefficients of the minimal polynomial

αν = cν (30)

then they can be determined by requiring that the L2 norm of the approximation ofthe solution in Krylov space is minimized

α = argminαν

∥∥∥∥∥Q(ν) +n∑ν=0

ανQ(ν)

∥∥∥∥∥2

(31)

The constraints for this optimization problem is obtained from the property of theminimal polynomial

n∑ν=0

αν = 1 (32)

Using the recursive property equation (19) and substituting it into equation (27),the following over-determined linear system results

n∑ν=0

ανQ(ν) = 0 (33)

Equations (32) and (33) represent a constrained optimization problem for findingthe extrapolation coefficients for the problem defined by equation (31). The result-ing system of normal equations is solved using a QR-decomposition algorithm.



Once the coefficients are determined, the solution is expressed by the followingextrapolation formula

Q(ν+1) = Q(ν) +n∑ν=0

ανQ(ν) (34)

Given the fact that equation (33) represent an over-determined system of linearequations, it is obvious that not all linear equations in that system will be satisfiedby the computed vector αν. This is the consequence of the inconsistent nature ofthe over-determined linear system in equation (33) and this fact plays an importantrole in the selection of the size of the dimension of the restart space.

In practical terms, the RRE algorithm is usually implemented in a restartedversion by collecting n vectors Q(ν) and storing them in the rectangular matrix[Q]

[Q] = [Q(0), Q(1), · · · , Q(n)] (35)

leading to the following matrix equation

[Q]α = 0 (36)

With previous definitions, the RRE algorithm is shown in Figure 12.1.It is evident from the RRE pseudo-code that the iterative algorithm acts as a

preconditioner for the RRE extrapolation algorithm and only the sequence of arraysQ(ν) is required in order to form the matrix [Q] that is used to obtain extrapola-tion coefficients. Therefore, the RRE algorithm is implemented as a wrapper around

Algorithm 1for k = 0, N

for ν = 0, nQ(ν) = −M(ν)(R(Q(ν)))Q(ν+1) = Q(ν) + Q(ν)

form matrix [Q]endFind α = argminαν ||Q(ν) + ∑n

ν=0 ανQ(ν)||2 such that∑nν=0 αν = 1

Q(k+1) = Q(k) + ∑nν=0 ανQ(ν)

if convergedstop

elserestart

end ifend

Figure 12.1. RRE pseudo-code.



existing code that implements the fixed-point algorithm. A further advantage of theproposed algorithm is that no explicit knowledge of the operator M(ν)(R(Q(ν)))in equation (11) is ever required; it is only required that the fixed-point algorithmin equation (10) produce a sequence of solutions. This particular property makesthis approach widely applicable and nonintrusive from the implementation pointof view. Furthermore, due to equivalence between Krylov spaces, equation (21),the RRE algorithm is equivalent to the GMRES method. In the particular case ofnonlinear flow solvers, the RRE algorithm corresponds to GMRES algorithm withnonlinear preconditioning and it is sometimes referred to as nonlinear GMRES.

Since the theory of RRE and its application relies on the linearized problem,most of the properties of the algorithm are valid within some small neighborhoodof the operator M(R(Q). Furthermore, since we are using a restarted version ofRRE, the choice of the restart space is very important. The size of the restart spaceis very important and somewhat problem dependent. Restarted versions of GMRESsuffer from the same problems [11] and there is no theory on how to choose thesize of the restart space. Therefore, some numerical experimentation is needed inorder to establish the proper size of the restart space for a given class of problems.

12.4 Numerical Experiments

As was shown in the algorithm in Figure 12.1, the RRE algorithm requires onlya sequence of arrays produced by the preconditioning solver M(R(Q(ν)). Dueto this property, RRE is implemented as a wrapper around several nonlinearflow solvers including coupled pressure-based, segregated pressure-based, explicitcoupled density-based and implicit coupled density-based solvers [19]. Here wedescribe results obtained through numerical experiments in the application of theRRE algorithm to mentioned solvers.

12.4.1 RRE acceleration of implicit density-based solver

Here we consider inviscid and viscous turbulent flow around NACA 0012 airfoil at0 angle of attack with the free-stream Mach number Ma = 0.7. The computationalgrid is shown in Figure 12.2 and it consists of 4, 800 finite volume cells. Thegoverning equations are described by two dimensional inviscid Euler equationsthat can be obtained from equation (1) by removing viscous fluxes Fv and droppingthe momentum, continuity, and energy term in the z-direction

∂

∂t

∫

Qd+∮∂

FcdA = 0 (37)

Here Q and Fc are given by

Q =

⎛⎜⎜⎝ρ

ρuρv

ρE

⎞⎟⎟⎠ , Fc =

⎛⎜⎜⎝ρV

ρuV + nxpρvV + nypρHV

⎞⎟⎟⎠



Grid

Figure 12.2. Computational grid for NACA 0012 airfoil.

The numerical flux scheme used to approximate fluxes at the centroids of facesof finite volumes is the Roe flux difference splitting scheme [1,6] with second-orderspatial accuracy. Steady state was obtained after 375 implicit iterations according tothe scheme in equation (8) and with CFL number equal to 25.A symmetric flow fieldpattern that was expected at 0 angle of attack is shown in Figure 12.3. Figure 12.4shows the convergence history of all conserved variables. Convergence behavioris overall linear, but a small marginally unstable mode in the momentum equationis observed that corresponds to almost a periodic half-sinusoidal oscillation of theresiduals. Results of the RRE algorithm that uses coupled density-based implicitsolver are shown in Figure 12.5. These results are obtained by using a size of therestart space of 25 and it is evident from Figure 12.5 that the accelerated caseconverges after 265 implicit iterations of density-based solver. Moreover, it can beobserved that an almost periodic marginally unstable mode evident in Figure 12.4is no longer present and it is substituted with higher more irregular modes. Theoverall savings in terms of number of iterations between baseline and acceleratedruns is about 100 iteration or 25%.

A more interesting case to consider is turbulent viscous flow over the sameNACA 0012 at an angle of attack of 1.25. For this angle of attack, a weak shockwave appears on the upper surface of the airfoil, close to the leading edge. The basicset of equations, equation (1) was extended with additional turbulence transportequations. The generic form of transport equations is given by equation (2) and in



Contours of Mach number May 20, 2007FLUENT 6.3 (2d, dp, dbns imp)

9.10e018.83e018.56e018.28e018.01e017.74e017.46e017.19e016.92e016.64e016.37e016.10e015.82e015.55e015.28e015.00e014.73e014.46e014.18e013.91e013.64e013.36e013.09e012.82e012.54e012.27e01

Figure 12.3. Contours of Mach number for NACA 0012 airfoil at the zero angleof attack using implicit density-based solver.

Coupled airfoil: No extrapolationscaled residuals May 20, 2007

FLUENT 6.3 (2d, dp, dbns imp)

Iterations

400350300250200150100500

1e02

1e00

1e02

1e04

1e06

1e08

1e10

1e12

1e14

energyy-velocityx-velocitycontinuityResiduals

Figure 12.4. Residual history of the baseline run (no acceleration) of NACA 0012airfoil in inviscid flow at the zero angle of attack using implicit density-based solver.



Coupled airfoil: With extrapolationscaled residuals May 20, 2007

FLUENT 6.3 (2d, dp, dbns imp)

Iterations

300250200150100500

1e02

1e00

1e02

1e04

1e06

1e08

1e10

1e12

1e14


Figure 12.5. Residual history of the the accelerated run of NACA 0012 airfoil ininviscid flow at the zero angle of attack using implicit density-basedsolver.

the case of two equation turbulence model it takes the following form:

∂

∂t

∫

d+∮∂

(F,c − F,v)dA −∫

Sd = 0

Here , F,c, and F,v are given by

=(ρKρε

), F,c =

(ρKVρε∗V

), F,v =

(nxτ

Kxx + nyτ

Kyy

nxτεxx + nyτ

εyy

)whereas S is given by

=(

P − ρε(Cε1fε1 − Cε2fε2ρε∗) ε

∗K + φε

)where P is production term [20] and τxx and τyy are normal turbulent viscousstresses [20]. The turbulence model that was used in the computation is realizablek − εmodel with the free stream turbulence conditions that correspond to 0.5% tur-bulence intensity with a viscous ratio of turbulent to laminar viscosity of 10. Theseconditions correspond to free stream conditions in the calm atmosphere. Results ofthe computations using baseline coupled implicit density-based solver are shown inFigure 12.6. Convergence behavior for this case is shown in Figure 12.7 where it canbe seen that the convergence was achieved after 390 nonlinear iterations. It is also



Contours of Mach number May 20, 2007FLUENT 6.3 (2d, dp, dbns imp, rke)

1.01e009.79e019.47e019.15e018.84e018.52e018.20e017.88e017.56e017.24e016.93e016.61e016.29e015.97e015.65e015.34e015.02e014.70e014.38e014.06e013.75e013.43e013.11e012.79e012.47e012.16e01

Figure 12.6. Mach contours of NACA 0012 airfoil in viscous turbulent flow at1.25 angle of attack using implicit density-based solver.

Coupled airfoil: No extrapolationscaled residuals May 20, 2007

FLUENT 6.3 (2d, dp, dbns imp, rke)

Iterations

400350300250200150100500

1e02

1e00

1e02

1e04

1e06

1e08

1e10

1e12

1e14

epsilonkenergyy-velocityx-velocitycontinuityResiduals

Figure 12.7. Residual history of the baseline (no acceleration) run of NACA 0012in viscous turbulent flow at 1.25 angle of attack using implicitdensity-based solver.



observed that turbulent dissipation rate experiences a plateau effect after 200 nonlin-ear iterations, keeping the level of residuals at 1E–07 without any further decrease.For practical computations, this level of residuals may be acceptable but it clearlyindicates a difficulty with the convergence of k-equation. Application of the RREaccelerating algorithm produces better convergence behavior since the problemconverges in 275 nonlinear iterations without the plateau effect in k-equation.

12.4.2 RRE acceleration of explicit density-based solver

Here again we consider inviscid and viscous turbulent flow around NACA 0012 air-foil at 0 and 1.25 angles of attack with the free-stream Mach number Ma = 0.7.However, this time we use the explicit multistage Runge–Kutta scheme with the fullapproximation storage scheme [1,18,19]. The same set of equations, equation (37)and the nonlinear update is performed by the explicit scheme in equation (7) withmodifications to accommodate FAS and Runge–Kutta multistage algorithms. Dueto stability restrictions, CFL number of 2 is used with second-order scheme in space(Figure 12.8).

The first case that was considered is the inviscid flow over NACA 0012 at the zeroangle of attack with free-stream Mach number Ma = 0.7. Convergence behavior ofresiduals of the conserved variables are shown in Figure 12.9 where it can be seenthat it takes approximately 3, 470 nonlinear iterations to reduce residuals belowtolerance level set to be 1E − 12.

Coupled airfoil: With extrapolationscaled residuals May 20, 2007

FLUENT 6.3 (2d, dp, dbns imp, rke)

Iterations

300250200150100500

1e02

1e00

1e02

1e04

1e06

1e08

1e10

1e12

1e14


Figure 12.8. Residual history of the accelerated run of NACA 0012 in viscousturbulent flow at 1.25 angle of attack using implicit density-basedsolver.



Explicit airfoil: No extrapolationscaled residuals May 22, 2007

FLUENT 6.3 (2d, dp, dbns exp)

Iterations

3500300025002000150010005000

1e02

1e00

1e02

1e04

1e06

1e08

1e10

1e12

1e14


Figure 12.9. Residual history of the baseline (no acceleration) run of NACA 0012in inviscid flow at the zero angle of attack using explicit density-basedsolver.

In the case of the RRE accelerated explicit density-based algorithm, the sizeof a restart space is 25 as in the case of implicit solvers. However, due to the factthat the explicit solvers are not very good preconditioners, the resulting Krylovsubspace does not have a good basis to guarantee convergence of the RRE method.It was determined experimentally that if every 10th correction vector is stored,then RRE converges. This behavior is induced by a very nonlinear behavior of thesystem on a scale of 10 to 15 iterations as it can be seen in Figure 12.9. Skippingiterations results in a better basis of the Krylov subspace and the RRE methodof accelerating explicit density-based solver converges as shown in Figure 12.10where it can be seen that the convergence is achieved after approximately 2, 250nonlinear iterations. This behavior requires further research and some possibleapproaches include evaluation of residuals to determine the good candidate for theextrapolation [14]. Some other measures of the quality of the Krylov subspace thatinvolve singular value decomposition of matrix [Q] and thresholding based onthe desired accuracy can be used.

A viscous turbulent case of NACA 0012 at an angle of attack of 1.25 withthe same free-stream conditions is examined next. Stalled convergence as shownin Figure 12.11 is experienced in this case with the residual levels in the vicinityof 1E − 06. Employing the same strategy of storing every 10th correction vec-tor, convergence is accelerated and stalled residuals are removed. Convergenceimprovement in this case is attributed to improved coupling between flow and tur-bulence equations by the RRE algorithm. Current algorithm in density-based solver



Explicit airfoil: With extrapolationscaled residuals May 22, 2007

FLUENT 6.3 (2d, dp, dbns exp)

Iterations

2,2502,0001,7501,5001,2501,0007505002500

1e02

1e00

1e02

1e04

1e06

1e08

1e10

1e12

1e14


Figure 12.10. Residual history of the accelerated run of NACA 0012 in inviscidflow at an 1.25 angle of attack using explicit density-based solver.

solves flow equations in a coupled manner and then it solves turbulence equationsseparately by using computed velocity field. This procedure sometimes leads intoconvergence problems that are observed in Figure 12.11 and the RRE algorithmhelps the convergence by providing the missing coupling between equations bycomputing one set of extrapolation coefficients used in extrapolation of all fields.Furthermore, in a view of previous connection between the RRE and GMRESmethod, this approach corresponds to applying the GMRES method on a coupledsystem of equations thus improving the convergence (Figure 12.12).

12.4.3 RRE acceleration of segregated pressure-based solver

In this section we consider acceleration of the computation of laminar incompress-ible flows by the RRE method. The basic set of equations is given in equation (1) andtheir specialization for the laminar incompressible flow is given by the followingequation: ∮

∂

(Fc − Fv) dA = 0 (38)

with the incompressibility constraint enforced through the divergence-freecondition ∫

∇ · (ρu)d = 0 (39)



Explicit airfoil: No extrapolationscaled residuals May 23, 2007

FLUENT 6.3 (2d, dp, dbns exp, ske)

Iterations

3,0002,5002,0001,5001,0005000

1e01

1e00

1e01

1e02

1e03

1e04

1e05

1e06

1e07


Figure 12.11. Residual history of the baseline (no acceleration) run of NACA 0012in a viscous turbulent flow at an 1.25 angle of attack using explicitdensity-based solver.

Explicit airfoil: With extrapolationscaled residuals May 23, 2007

FLUENT 6.3 (2d, dp, dbns exp, ske)

Iterations

2,5002,2502,0001,7501,5001,2501,0007505002500

1e02

1e00

1e02

1e04

1e06

1e08

1e10

1e12

1e14


Figure 12.12. Residual history of the accelerated run of NACA 0012 in a viscousturbulent flow at an 1.25 angle of attack using explicit density-basedsolver.



Z

Y

X

Grid Feb 04, 2007FLUENT 6.3 (3d, dp, pbns, lam)

Figure 12.13. Computational grid for laminar viscous flow in serpentine geometry.

Variables in equation (1) for the case of incompressible laminar flows are given byfollowing expressions:

Fc =⎛⎝ ρuV + nxpρvV + nypρwV + nzp

⎞⎠ , Fv =⎛⎝nxτxx + nyτxy + nzτxz

nxτyx + nyτyy + nzτyznxτzx + nyτzy + nzτzz

⎞⎠The incompressible system of equations (38) and (39) is usually solved through

the projection methods [1,3,4] where the primary unknown variables are the prim-itive variables such as pressure and velocity. Furthermore, due to the kinematicconstraint, equation (39) imposed on the velocity field, there is no direct linkbetween pressure and velocity fields, a special pressure equation is usually formed[1,4]. Forming the pressure equation, together with the momentum equation andthe boundary conditions, creates a well-posed problem that can be solved in a cou-pled or segregated manner. If the SIMPLE algorithm [4] is adopted, segregatedalgorithm results.

As a study case, we consider the flow through the geometry shown inFigure 12.13. Due to the nonlinearities in equation (38), and due to the initialguesses that lay out of the convergence domain of the SIMPLE methods, relaxationfactors are used in the discrete system of equations to maintain the stability ofthe computations. Standard relaxation factors (0.3 pressure and 0.7 for momentumequation) together with the second-order formulations for pressure and velocityvariables is used in computation. The problem in Figure 12.13 is described by one



Serpentine: No extrapolationvelocity vectors colored by velocity magnitude (m/s) May 23, 2007

FLUENT 6.3 (3d, dp, pbns, lam)

2.24e00

2.13e00

2.02e00

1.91e00

1.79e00

1.68e00

1.57e00

1.46e00

1.35e00

1.24e00

1.13e00

1.02e00

9.05e01

7.93e01

6.82e01

5.71e01

4.60e01

3.48e01

2.37e01

1.26e01

1.48e02Z

Y

X

Figure 12.14. Vector field in incompressible flow plotted in the middle plane of theserpentine geometry.

inflow boundary condition prescribed as a velocity inlet, one outlet boundary con-dition prescribed as an uniform pressure outlet, and no-slip walls. Problem wasdiscretized using hexagonal 8, 900 finite volumes. System of discretized equationsfor the pressure and velocity variables was solved using implicit formulation thatforms four systems of linear equations that are solved using the AMG method[18,19] in a segregated manner and velocity vectors on the middle plane are shownin Figure 12.14. Convergence criterion was set for all equations to be 1E − 14 forthe norm of residuals and the results of the computations are shown in Figure 12.15where it can be seen that it takes close to 2, 500 iterations to reach the prescribedlevel of convergence. The accelerated case is shown in Figure 12.16 where theRRE algorithm results in approximately 1, 000 iterations required to reach the pre-scribed levels of residuals. As in the case of explicit density-based solver, the sizeof a restart space is 25 and RRE stored every 10th vector in order to obtain usefulbasis of Krylov subspace.

12.4.4 RRE acceleration of coupled pressure-based solver

The set of governing equations (38) and (39) for the incompressible flow can be dis-cretized and solved simultaneously resulting in a so-called coupled pressure-basedsolver. In contrast to the segregated solver, matrix coefficients obtained throughcoupled discretization result in a point coupled system where three velocities and



Serpentine: No extrapolationscaled residuals May 23, 2007


Iterations

4,0003,5003,0002,5002,0001,5001,0005000

1e02

1e00

1e02

1e04

1e06

1e08

1e10

1e12

1e14

1e16

z-velocityy-velocityx-velocitycontinuityResiduals

Figure 12.15. Residual history of the baseline run (no acceleration) of the incom-pressible laminar flow in the serpentine geometry using segregatedpressure-based solver.

Serpentine: With extrapolationscaled residuals

May 23, 2007FLUENT 6.3 (3d, dp, pbns, lam)

Z

Y

XIterations

4,0003,5003,0002,5002,0001,5001,0005000

1e02

1e00

1e02

1e04

1e06

1e08

1e10

1e12

1e14

1e16


Figure 12.16. Residual history of the accelerated run of the incompressible laminarflow in the serpentine geometry using segregated pressure-basedsolver.



Z

Y

X

SphereGrid


Figure 12.17. Computational grid in the middle plane of a sphere in a free streamflow at Re = 250.

one pressure is stored in a block coefficient for each finite volume cell [19]. Thissystem of equations is solved using again the AMG method with the modificationsfor block coefficients.

The RRE method of acceleration was applied to the case of sphere in a freestream flow conditions with Reynolds number equal to 250. The grid consists of1.2E + 06 finite volume cells of a tetrahedral shape as shown in Figure 12.17.Stability of the computational run was controlled by the CFL number which waschosen to be 200 in this case.The flow is laminar and the contours of the pressure andvectors of the velocity field are shown in Figures 12.18 and 12.19, respectively. Theconvergence history of baseline run without acceleration is shown in Figure 12.20where the slow convergence is observed. Convergence behavior for the acceleratedcase is shown in Figure 12.21 where faster convergence is observed. As in the caseof density-based implicit solver, restart base is selected to be 25 and correctionvectors are collected after each nonlinear iteration.

12.5 Conclusion

We have demonstrated the application of the RRE method as an acceleration methodfor the nonlinear solvers in CFD. The RRE method relies on the recursive propertyof the fixed-point algorithms used in CFD to form the basis of the Krylov subspacethat contains the solution to the problem. Due to the nonlinearity of the fixed-point



PBCS Sphere: With extrapolationcontours of static pressure (pascal) May 24, 2007


4.98e01

4.59e01

4.19e01

3.79e01

3.40e01

3.00e01

2.60e01

2.21e01

1.81e01

1.41e01

1.02e01

6.21e02

2.24e02

1.72e02

5.69e02

9.66e02

1.36e01

1.76e01

2.16e01

2.55e01

2.95e01Z

YX

Figure 12.18. Pressure field in the middle plane of a sphere in a free stream flowat Re = 250 using coupled pressure-based solver.

PBCS sphere: With extrapolationvelocity vectors colored by velocity magnitude (m/s)


1.18e00

1.12e00

1.06e00

1.00e00

9.46e01

8.87e01

8.28e01

7.69e01

7.10e01

6.51e01

5.92e01

5.33e01

4.74e01

4.15e01

3.56e01

2.97e01

2.37e01

1.78e01

1.19e01

6.04e02

1.33e03 Z

Y X

Figure 12.19. Velocity vectors in the middle plane of a sphere in a free stream flowat Re = 250 using coupled pressure-based solver.



PBCS sphere: No extrapolationscaled residuals



Iterations

2502252001751501251007550250

1e01

1e00

1e01

1e02

1e03

1e04

1e05

1e06

1e07

1e08

1e09

Figure 12.20. Residual history of the baseline (no acceleration) run of a sphere ina free stream flow at Re = 250 using coupled pressure-based solver.

PBCS sphere: With extrapolationscaled residuals


Iterations

2502252001751501251007550250

1e02

1e00

1e02

1e04

1e06

1e08

1e10

1e12

1e14

1e16

1e18


Figure 12.21. Residual history of the accelerated run of a sphere in a free streamflow at Re = 250 using coupled pressure-based solver.



function of CFD algorithms, restarted algorithm is used to make the algorithmmemory efficient and to rely on local linear properties of the restarted Krylovsubspace to represent the solution correctly. The proposed algorithm is shown tobe nonintrusive and applicable to a wide range of solvers including explicit andimplicit formulations of pressure- and density-based algorithms. The RRE algo-rithm requires only the storage of correction vectors and does not require evaluationof the residuals which makes this algorithm easy to implement for the solvers thatdo not form residual vector. Improved coupling of the systems of equations whenused with the RRE algorithm is demonstrated and shown to improve convergenceproperties of segregated solvers.

References

[1] Wesseling, P. Principles of Computational Fluid Dynamics, Springer-Verlag, NewYork, 2000.

[2] Patankar, S. V. Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York,1980.

[3] Chorin, A. J. A Numerical Method for Solving Incompressible Viscous FlowProblems, Journal of Computational Physics, 2(1), pp. 12–26, 1967.

[4] Ferziger, J. H., and Peric, M. Computational Methods for Fluid Dynamics, Springer,Berlin, 1996.

[5] Hirsch, C. Numerical Computation of Internal and External Flows, 2, John Wiley,Chichester, 1990.

[6] Laney, C. B. Computational Gasdynamics, Cambridge University Press, Cambridge,1998.

[7] Allgower, E. L., and Georg, K. Introduction to Numerical Continuation Methods,SIAM, Philadelphia, 1987.

[8] Deuflhard, P. Newton Methods for Nonlinear Problems, Springer-Verlag, Berlin, 2004.[9] Wilcox, D. C. Turbulence Modeling for CFD, DCW Industries, 2006.

[10] Brezinski, C., and Zaglia, M. R. Extrapolation Methods Theory and Practice, ElsevierScience Publishing, Amsterdam, 1991.

[11] Saad, Y. Iterative Methods for Sparse Linear Systems SIAM, Philadelphia, 2004.[12] Jameson, A., Schmidt, W., and Turkel, E. Numerical Solutions of the Euler Equations

by Finite Volume Methods Using Runge-Kutta Time Stepping, AIAA Paper 81–1259,1981.

[13] Sidi, A., and Celestina, M. L. Convergence Acceleration for Vector Sequences andApplication to Computational Fluid Dynamics, NASA Technical Memolen, 101327(ICOMP-88-17).

[14] Washio, T., and Oosterlee, C. W. Krylov Subspace Acceleration for Nonlinear Multi-grid Schemes, Electronic Transactions on Numerical Analysis, 6, pp. 271–290,December 1997, ISSN 1068-9613. Institute for Computational Mechanics in Propul-sion, Lewis Research Center, Clevland, OH.

[15] Jaschke, L. Preconditioned Arnoldi Methods for Systems of Nonlinear Equations,PhD Thesis, Swiss Federal Institute of Technology, Zurich, Switzerland, 2003.

[16] Andersson, C. Solving Linear Equations on Parallel Distributed MemoryArchitecturesby Extrapolation, Technical Report, TRITA-NA-E9746, Department of NumericalAnalysis and Computer Science, Royal Institute of Technology, Sweden, 1997.



[17] Jemcov, A., Maruszewski, J. P., and Jasak, H. Performance Improvement of AlgebraicMutligrid Solver by Vector Sequence Extrpolation, 15th Annual Conference of CFDSociety of Canada, May 27–31, Toronto, 2007.

[18] Briggs, W., Henson, V. E., and McCormick, S. A Mutligrid Tutorial, SIAM,Philadelphia, 2000.

[19] Fluent Inc., Fluent Users Manual, Fluent Inc., 2006.[20] Pope, S. B. Turbulent Flows, Cambridge University Press, Cambridge, 2000.

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